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Properties

Label 4-10e2-1.1-c25e2-0-1
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $1568.13$
Root an. cond. $6.29282$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.19e3·2-s − 9.70e4·3-s + 5.03e7·4-s − 4.88e8·5-s − 7.95e8·6-s − 1.61e10·7-s + 2.74e11·8-s − 1.24e12·9-s − 4.00e12·10-s + 1.44e13·11-s − 4.88e12·12-s + 9.34e13·13-s − 1.32e14·14-s + 4.74e13·15-s + 1.40e15·16-s + 4.99e14·17-s − 1.01e16·18-s − 3.83e15·19-s − 2.45e16·20-s + 1.56e15·21-s + 1.18e17·22-s + 1.69e17·23-s − 2.66e16·24-s + 1.78e17·25-s + 7.65e17·26-s + 1.59e17·27-s − 8.12e17·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.105·3-s + 3/2·4-s − 0.894·5-s − 0.149·6-s − 0.440·7-s + 1.41·8-s − 1.46·9-s − 1.26·10-s + 1.38·11-s − 0.158·12-s + 1.11·13-s − 0.623·14-s + 0.0943·15-s + 5/4·16-s + 0.207·17-s − 2.07·18-s − 0.397·19-s − 1.34·20-s + 0.0464·21-s + 1.95·22-s + 1.61·23-s − 0.149·24-s + 3/5·25-s + 1.57·26-s + 0.205·27-s − 0.660·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1568.13\)
Root analytic conductor: \(6.29282\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(6.958534474\)
\(L(\frac12)\) \(\approx\) \(6.958534474\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p^{12} T )^{2} \)
5$C_1$ \( ( 1 + p^{12} T )^{2} \)
good3$D_{4}$ \( 1 + 3596 p^{3} T + 1717509338 p^{6} T^{2} + 3596 p^{28} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 16137160124 T - 7807648962814970958 p^{2} T^{2} + 16137160124 p^{25} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 - 14416600801344 T + \)\(18\!\cdots\!66\)\( p^{2} T^{2} - 14416600801344 p^{25} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 - 93404548866628 T + \)\(11\!\cdots\!14\)\( p T^{2} - 93404548866628 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 - 29368640128308 p T + \)\(34\!\cdots\!42\)\( p^{2} T^{2} - 29368640128308 p^{26} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 + 201589715363480 p T + \)\(39\!\cdots\!18\)\( p^{2} T^{2} + 201589715363480 p^{26} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 - 169271186050723788 T + \)\(11\!\cdots\!14\)\( p T^{2} - 169271186050723788 p^{25} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 - 569443622853716940 T + \)\(69\!\cdots\!98\)\( T^{2} - 569443622853716940 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 - 3982769593868414584 T + \)\(41\!\cdots\!66\)\( T^{2} - 3982769593868414584 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 - 28241444663789328196 T - \)\(14\!\cdots\!82\)\( T^{2} - 28241444663789328196 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 - 36676428854020432524 T + \)\(35\!\cdots\!46\)\( T^{2} - 36676428854020432524 p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - \)\(90\!\cdots\!28\)\( T + \)\(34\!\cdots\!82\)\( T^{2} - \)\(90\!\cdots\!28\)\( p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 - \)\(95\!\cdots\!96\)\( T + \)\(42\!\cdots\!18\)\( T^{2} - \)\(95\!\cdots\!96\)\( p^{25} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 - \)\(27\!\cdots\!88\)\( T + \)\(16\!\cdots\!22\)\( T^{2} - \)\(27\!\cdots\!88\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 - \)\(45\!\cdots\!80\)\( T - \)\(18\!\cdots\!02\)\( T^{2} - \)\(45\!\cdots\!80\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 + \)\(26\!\cdots\!36\)\( T + \)\(93\!\cdots\!26\)\( T^{2} + \)\(26\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 + \)\(14\!\cdots\!24\)\( T + \)\(65\!\cdots\!58\)\( T^{2} + \)\(14\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 + \)\(17\!\cdots\!36\)\( T + \)\(34\!\cdots\!26\)\( T^{2} + \)\(17\!\cdots\!36\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 + \)\(34\!\cdots\!32\)\( T + \)\(83\!\cdots\!42\)\( T^{2} + \)\(34\!\cdots\!32\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 + \)\(92\!\cdots\!00\)\( T + \)\(76\!\cdots\!98\)\( T^{2} + \)\(92\!\cdots\!00\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 - \)\(68\!\cdots\!48\)\( T - \)\(11\!\cdots\!38\)\( T^{2} - \)\(68\!\cdots\!48\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 - \)\(38\!\cdots\!80\)\( T + \)\(14\!\cdots\!98\)\( T^{2} - \)\(38\!\cdots\!80\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 + \)\(16\!\cdots\!64\)\( T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(16\!\cdots\!64\)\( p^{25} T^{3} + p^{50} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.87198458028148213054408099432, −14.56420477160205827152756985243, −13.79238779399364083481388158639, −13.17722026568140347982586287076, −12.10701551749495798808679016525, −12.00045798559841793275080789727, −10.92490040001171314166845916862, −10.90507687333009033194744043577, −9.150923791724951314718736346031, −8.674934187930878223447820595810, −7.59939937718354106091567344807, −6.77611697309666041293193480564, −6.03026217565657804221099389387, −5.61075416557292730241010202054, −4.20830786797002826222917943676, −4.13850937697760931376022353106, −2.94693380386204346244762430629, −2.79622925444236206197409075218, −1.23991409486712841662795472706, −0.67250797503421422148198550273, 0.67250797503421422148198550273, 1.23991409486712841662795472706, 2.79622925444236206197409075218, 2.94693380386204346244762430629, 4.13850937697760931376022353106, 4.20830786797002826222917943676, 5.61075416557292730241010202054, 6.03026217565657804221099389387, 6.77611697309666041293193480564, 7.59939937718354106091567344807, 8.674934187930878223447820595810, 9.150923791724951314718736346031, 10.90507687333009033194744043577, 10.92490040001171314166845916862, 12.00045798559841793275080789727, 12.10701551749495798808679016525, 13.17722026568140347982586287076, 13.79238779399364083481388158639, 14.56420477160205827152756985243, 14.87198458028148213054408099432

Graph of the $Z$-function along the critical line