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Properties

Label 4-65e2-1.1-c1e2-0-2
Degree $4$
Conductor $4225$
Sign $1$
Analytic cond. $0.269389$
Root an. cond. $0.720435$
Motivic weight $1$
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  + 2·3-s − 4-s − 2·5-s + 4·7-s − 6·11-s − 2·12-s + 2·13-s − 4·15-s − 3·16-s − 2·19-s + 2·20-s + 8·21-s + 6·23-s + 3·25-s − 2·27-s − 4·28-s − 12·29-s + 10·31-s − 12·33-s − 8·35-s − 8·37-s + 4·39-s + 10·43-s + 6·44-s + 12·47-s − 6·48-s − 2·49-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.51·7-s − 1.80·11-s − 0.577·12-s + 0.554·13-s − 1.03·15-s − 3/4·16-s − 0.458·19-s + 0.447·20-s + 1.74·21-s + 1.25·23-s + 3/5·25-s − 0.384·27-s − 0.755·28-s − 2.22·29-s + 1.79·31-s − 2.08·33-s − 1.35·35-s − 1.31·37-s + 0.640·39-s + 1.52·43-s + 0.904·44-s + 1.75·47-s − 0.866·48-s − 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.269389\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9122508780\)
\(L(\frac12)\) \(\approx\) \(0.9122508780\)
\(L(\frac{3}{2})\) 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)$
bad5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T - 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−15.11767240512088034068912454226, −14.66513874748013962570787213521, −13.92119611272951525106243315187, −13.85625533186125509599468889503, −12.99851871291267216655788274607, −12.69940336201952227391431771443, −11.71547383331981146360180340762, −11.04697580980788585086419591350, −10.97194024744893244170118313479, −10.08581651493428023688237358708, −9.036330438579938888990771144807, −8.637227620885383269419080311816, −8.332943819541073368415088174114, −7.58651044449099949588254922570, −7.30852618143992369946232833200, −5.80996823928595783242142273545, −4.93498568789334957600638770142, −4.42295819222663121590160150462, −3.30278022896545756679526281316, −2.33316184409005000205408342494, 2.33316184409005000205408342494, 3.30278022896545756679526281316, 4.42295819222663121590160150462, 4.93498568789334957600638770142, 5.80996823928595783242142273545, 7.30852618143992369946232833200, 7.58651044449099949588254922570, 8.332943819541073368415088174114, 8.637227620885383269419080311816, 9.036330438579938888990771144807, 10.08581651493428023688237358708, 10.97194024744893244170118313479, 11.04697580980788585086419591350, 11.71547383331981146360180340762, 12.69940336201952227391431771443, 12.99851871291267216655788274607, 13.85625533186125509599468889503, 13.92119611272951525106243315187, 14.66513874748013962570787213521, 15.11767240512088034068912454226

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