↑

Properties

Label 4-10e2-1.1-c21e2-0-0
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $781.075$
Root an. cond. $5.28656$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.04e3·2-s + 1.00e5·3-s + 3.14e6·4-s + 1.95e7·5-s − 2.05e8·6-s + 1.32e9·7-s − 4.29e9·8-s + 7.55e9·9-s − 4.00e10·10-s + 2.18e10·11-s + 3.15e11·12-s − 4.12e11·13-s − 2.72e12·14-s + 1.95e12·15-s + 5.49e12·16-s − 1.10e13·17-s − 1.54e13·18-s − 5.20e13·19-s + 6.14e13·20-s + 1.33e14·21-s − 4.47e13·22-s + 8.09e13·23-s − 4.30e14·24-s + 2.86e14·25-s + 8.43e14·26-s + 1.55e15·27-s + 4.18e15·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.980·3-s + 3/2·4-s + 0.894·5-s − 1.38·6-s + 1.77·7-s − 1.41·8-s + 0.721·9-s − 1.26·10-s + 0.254·11-s + 1.47·12-s − 0.829·13-s − 2.51·14-s + 0.877·15-s + 5/4·16-s − 1.33·17-s − 1.02·18-s − 1.94·19-s + 1.34·20-s + 1.74·21-s − 0.359·22-s + 0.407·23-s − 1.38·24-s + 3/5·25-s + 1.17·26-s + 1.45·27-s + 2.66·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(781.075\)
Root analytic conductor: \(5.28656\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(3.065780323\)
\(L(\frac12)\) \(\approx\) \(3.065780323\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 + p^{10} T )^{2} \)
5$C_1$ \( ( 1 - p^{10} T )^{2} \)
good3$D_{4}$ \( 1 - 33436 p T + 10326754 p^{5} T^{2} - 33436 p^{22} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 - 189842188 p T + 29983126517490222 p^{2} T^{2} - 189842188 p^{22} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 - 21869194224 T + \)\(12\!\cdots\!66\)\( T^{2} - 21869194224 p^{21} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 + 412060138772 T + \)\(36\!\cdots\!94\)\( p T^{2} + 412060138772 p^{21} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 + 11074763456364 T + \)\(79\!\cdots\!74\)\( p T^{2} + 11074763456364 p^{21} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 + 52072861011560 T + \)\(91\!\cdots\!02\)\( p T^{2} + 52072861011560 p^{21} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 - 80988875151948 T + \)\(37\!\cdots\!22\)\( T^{2} - 80988875151948 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 - 264315081456060 T + \)\(28\!\cdots\!58\)\( T^{2} - 264315081456060 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 - 2126016880188664 T + \)\(36\!\cdots\!86\)\( T^{2} - 2126016880188664 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 + 9721711577644724 T + \)\(16\!\cdots\!18\)\( T^{2} + 9721711577644724 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 - 221161031209870884 T + \)\(25\!\cdots\!46\)\( T^{2} - 221161031209870884 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 + 43823174002198412 T + \)\(40\!\cdots\!22\)\( T^{2} + 43823174002198412 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 - 4384129302297468 p T + \)\(22\!\cdots\!98\)\( T^{2} - 4384129302297468 p^{22} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 + 612435371018710692 T + \)\(25\!\cdots\!22\)\( T^{2} + 612435371018710692 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 - 4752208294289856120 T + \)\(41\!\cdots\!02\)\( p T^{2} - 4752208294289856120 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 - 4748119038295687324 T + \)\(61\!\cdots\!66\)\( T^{2} - 4748119038295687324 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 - 22301925713082765436 T + \)\(56\!\cdots\!58\)\( T^{2} - 22301925713082765436 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 - 2201035283486261544 T + \)\(49\!\cdots\!26\)\( T^{2} - 2201035283486261544 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 - 56149494460474652548 T + \)\(34\!\cdots\!22\)\( T^{2} - 56149494460474652548 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 + 42382706176352134640 T + \)\(93\!\cdots\!58\)\( T^{2} + 42382706176352134640 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 - 1335343926003173268 T - \)\(64\!\cdots\!78\)\( T^{2} - 1335343926003173268 p^{21} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 - \)\(25\!\cdots\!80\)\( T + \)\(18\!\cdots\!78\)\( T^{2} - \)\(25\!\cdots\!80\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 + \)\(15\!\cdots\!04\)\( T + \)\(16\!\cdots\!98\)\( T^{2} + \)\(15\!\cdots\!04\)\( p^{21} T^{3} + p^{42} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.06818046620703130116479767172, −15.21912155159924401248132221141, −14.49413679433034969440174975376, −14.35683781409278282128308605557, −13.11934151912874543148122010925, −12.35154341522367395879062644715, −11.10058017522273584326129807660, −10.83923921608156252125173020419, −9.876867116057108418201930643452, −9.124673078026105705924920854388, −8.489474929124001929838925423219, −8.101501770434156580374706528036, −7.06369318748437245070050587819, −6.38797998370977605118697145051, −5.00712838776311479702800748208, −4.21010937299549460776127580880, −2.38643130931047642659572170273, −2.35774451691357030838514304202, −1.52849586582870041598465895635, −0.67386207780214672162600015047, 0.67386207780214672162600015047, 1.52849586582870041598465895635, 2.35774451691357030838514304202, 2.38643130931047642659572170273, 4.21010937299549460776127580880, 5.00712838776311479702800748208, 6.38797998370977605118697145051, 7.06369318748437245070050587819, 8.101501770434156580374706528036, 8.489474929124001929838925423219, 9.124673078026105705924920854388, 9.876867116057108418201930643452, 10.83923921608156252125173020419, 11.10058017522273584326129807660, 12.35154341522367395879062644715, 13.11934151912874543148122010925, 14.35683781409278282128308605557, 14.49413679433034969440174975376, 15.21912155159924401248132221141, 16.06818046620703130116479767172

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