Properties

Label 4-70e2-1.1-c1e2-0-2
Degree $4$
Conductor $4900$
Sign $1$
Analytic cond. $0.312428$
Root an. cond. $0.747631$
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-s − 3-s + 5-s − 6-s − 7-s − 8-s + 3·9-s + 10-s + 6·11-s − 8·13-s − 14-s − 15-s − 16-s + 3·18-s − 2·19-s + 21-s + 6·22-s + 3·23-s + 24-s − 8·26-s − 8·27-s − 6·29-s − 30-s − 8·31-s − 6·33-s − 35-s + 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 9-s + 0.316·10-s + 1.80·11-s − 2.21·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s + 0.707·18-s − 0.458·19-s + 0.218·21-s + 1.27·22-s + 0.625·23-s + 0.204·24-s − 1.56·26-s − 1.53·27-s − 1.11·29-s − 0.182·30-s − 1.43·31-s − 1.04·33-s − 0.169·35-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.312428\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{70} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9876589257\)
\(L(\frac12)\) \(\approx\) \(0.9876589257\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−14.81856775359933444898427806420, −14.64560738992494087806062720888, −13.95405510801181231302130060545, −13.10628249052049853098496413897, −12.86032123579720894901900042944, −12.43727438283056963935294979956, −11.66137905510734289213836183759, −11.46184298868531629539847871046, −10.50931177147680436079438206836, −9.836512944089622846365789091494, −9.260788071972860386282139750579, −9.192176858265344691468648731669, −7.64235446639417735456485637220, −7.20871338231226987004533945840, −6.51434725552277554544789032162, −5.87268498267100909281846045227, −5.05512227067677025191579081551, −4.36637257403942965751068337280, −3.60004486490227968671810948230, −2.06539597472512688526896396217, 2.06539597472512688526896396217, 3.60004486490227968671810948230, 4.36637257403942965751068337280, 5.05512227067677025191579081551, 5.87268498267100909281846045227, 6.51434725552277554544789032162, 7.20871338231226987004533945840, 7.64235446639417735456485637220, 9.192176858265344691468648731669, 9.260788071972860386282139750579, 9.836512944089622846365789091494, 10.50931177147680436079438206836, 11.46184298868531629539847871046, 11.66137905510734289213836183759, 12.43727438283056963935294979956, 12.86032123579720894901900042944, 13.10628249052049853098496413897, 13.95405510801181231302130060545, 14.64560738992494087806062720888, 14.81856775359933444898427806420

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