Properties

Degree 4
Conductor $ 2^{2} \cdot 5^{2} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 10·5-s + 50·9-s − 56·11-s + 16·16-s + 120·19-s + 40·20-s − 25·25-s − 180·29-s − 256·31-s − 200·36-s + 484·41-s + 224·44-s − 500·45-s + 10·49-s + 560·55-s + 40·59-s + 1.08e3·61-s − 64·64-s − 2.25e3·71-s − 480·76-s + 1.44e3·79-s − 160·80-s + 1.77e3·81-s + 980·89-s − 1.20e3·95-s − 2.80e3·99-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 1.85·9-s − 1.53·11-s + 1/4·16-s + 1.44·19-s + 0.447·20-s − 1/5·25-s − 1.15·29-s − 1.48·31-s − 0.925·36-s + 1.84·41-s + 0.767·44-s − 1.65·45-s + 0.0291·49-s + 1.37·55-s + 0.0882·59-s + 2.27·61-s − 1/8·64-s − 3.77·71-s − 0.724·76-s + 2.05·79-s − 0.223·80-s + 2.42·81-s + 1.16·89-s − 1.29·95-s − 2.84·99-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{10} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 100,\ (\ :3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $0.664284$
$L(\frac12)$  $\approx$  $0.664284$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 4. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p^{2} T^{2} \)
5$C_2$ \( 1 + 2 p T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 - 50 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 28 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4250 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 5730 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 60 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 20970 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 90 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 128 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 45610 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 242 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 27970 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 156570 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 286090 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 20 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 542 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 413170 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 1128 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 378610 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 720 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 915090 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 490 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 294590 T^{2} + p^{6} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−20.75404921372514853975775707566, −20.49393296500168893853370391538, −19.24234170159821737585396352626, −18.93994862899296652043141094959, −18.00134541967902288543044622023, −17.98633281071982953329448072587, −16.21971445973795486680043857101, −16.12713599752384189961533448556, −15.36501571992824630772995142126, −14.56566335222668874528233179028, −13.24272907549937557694067291051, −13.04587947367273779802550776539, −12.08341892017131722999543459693, −10.98819281810985439264008946563, −10.10423520548203916488377750395, −9.264129044356701969990396274366, −7.68917674492804863897978674182, −7.43215939908228079028033812646, −5.30666260524625395791180327754, −3.96135615060000715278264354271, 3.96135615060000715278264354271, 5.30666260524625395791180327754, 7.43215939908228079028033812646, 7.68917674492804863897978674182, 9.264129044356701969990396274366, 10.10423520548203916488377750395, 10.98819281810985439264008946563, 12.08341892017131722999543459693, 13.04587947367273779802550776539, 13.24272907549937557694067291051, 14.56566335222668874528233179028, 15.36501571992824630772995142126, 16.12713599752384189961533448556, 16.21971445973795486680043857101, 17.98633281071982953329448072587, 18.00134541967902288543044622023, 18.93994862899296652043141094959, 19.24234170159821737585396352626, 20.49393296500168893853370391538, 20.75404921372514853975775707566

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