Properties

Degree 2
Conductor $ 5^{2} \cdot 241 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s + 2.50·3-s + 5.29·4-s − 6.77·6-s − 0.354·7-s − 8.89·8-s + 3.29·9-s + 4.18·11-s + 13.2·12-s + 3.72·13-s + 0.958·14-s + 13.4·16-s + 6.46·17-s − 8.88·18-s − 1.31·19-s − 0.890·21-s − 11.3·22-s + 4.10·23-s − 22.3·24-s − 10.0·26-s + 0.728·27-s − 1.87·28-s − 8.85·29-s + 5.11·31-s − 18.4·32-s + 10.5·33-s − 17.4·34-s + ⋯
L(s)  = 1  − 1.90·2-s + 1.44·3-s + 2.64·4-s − 2.76·6-s − 0.134·7-s − 3.14·8-s + 1.09·9-s + 1.26·11-s + 3.83·12-s + 1.03·13-s + 0.256·14-s + 3.35·16-s + 1.56·17-s − 2.09·18-s − 0.300·19-s − 0.194·21-s − 2.41·22-s + 0.856·23-s − 4.55·24-s − 1.97·26-s + 0.140·27-s − 0.355·28-s − 1.64·29-s + 0.918·31-s − 3.26·32-s + 1.82·33-s − 2.99·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6025\)    =    \(5^{2} \cdot 241\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6025} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.820061803$
$L(\frac12)$  $\approx$  $1.820061803$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;241\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;241\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 2.70T + 2T^{2} \)
3 \( 1 - 2.50T + 3T^{2} \)
7 \( 1 + 0.354T + 7T^{2} \)
11 \( 1 - 4.18T + 11T^{2} \)
13 \( 1 - 3.72T + 13T^{2} \)
17 \( 1 - 6.46T + 17T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
23 \( 1 - 4.10T + 23T^{2} \)
29 \( 1 + 8.85T + 29T^{2} \)
31 \( 1 - 5.11T + 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 0.673T + 43T^{2} \)
47 \( 1 + 5.22T + 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 - 4.04T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 7.76T + 73T^{2} \)
79 \( 1 + 1.17T + 79T^{2} \)
83 \( 1 + 6.25T + 83T^{2} \)
89 \( 1 - 3.80T + 89T^{2} \)
97 \( 1 - 9.91T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.179763805116092928151195459172, −7.76458729862475505946014067873, −7.02077981730225841302149636073, −6.40566406270077212170620869456, −5.56310808258554733405527658971, −3.83074232210901338267723503995, −3.40044102133817410058150930629, −2.51064488947241900811206141122, −1.60136278667572095260678791810, −0.963880784943121677122008610201, 0.963880784943121677122008610201, 1.60136278667572095260678791810, 2.51064488947241900811206141122, 3.40044102133817410058150930629, 3.83074232210901338267723503995, 5.56310808258554733405527658971, 6.40566406270077212170620869456, 7.02077981730225841302149636073, 7.76458729862475505946014067873, 8.179763805116092928151195459172

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