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  + 0.342·2-s − 2.18·3-s − 1.88·4-s − 0.747·6-s − 1.82·7-s − 1.32·8-s + 1.77·9-s + 5.99·11-s + 4.11·12-s − 3.70·13-s − 0.624·14-s + 3.31·16-s + 1.64·17-s + 0.607·18-s − 3.15·19-s + 3.98·21-s + 2.05·22-s + 5.46·23-s + 2.90·24-s − 1.26·26-s + 2.67·27-s + 3.43·28-s + 7.24·29-s − 9.41·31-s + 3.79·32-s − 13.0·33-s + 0.562·34-s + ⋯
L(s)  = 1  + 0.241·2-s − 1.26·3-s − 0.941·4-s − 0.305·6-s − 0.689·7-s − 0.469·8-s + 0.591·9-s + 1.80·11-s + 1.18·12-s − 1.02·13-s − 0.166·14-s + 0.827·16-s + 0.398·17-s + 0.143·18-s − 0.724·19-s + 0.870·21-s + 0.437·22-s + 1.13·23-s + 0.592·24-s − 0.248·26-s + 0.515·27-s + 0.649·28-s + 1.34·29-s − 1.69·31-s + 0.669·32-s − 2.27·33-s + 0.0964·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6277378725$
$L(\frac12)$  $\approx$  $0.6277378725$
$L(\frac{3}{2})$   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 \( 1 - 0.342T + 2T^{2} \)
3 \( 1 + 2.18T + 3T^{2} \)
7 \( 1 + 1.82T + 7T^{2} \)
11 \( 1 - 5.99T + 11T^{2} \)
13 \( 1 + 3.70T + 13T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 + 3.15T + 19T^{2} \)
23 \( 1 - 5.46T + 23T^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
31 \( 1 + 9.41T + 31T^{2} \)
37 \( 1 + 1.27T + 37T^{2} \)
41 \( 1 + 5.81T + 41T^{2} \)
43 \( 1 + 7.82T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 + 8.81T + 53T^{2} \)
59 \( 1 + 7.78T + 59T^{2} \)
61 \( 1 - 1.03T + 61T^{2} \)
67 \( 1 - 8.39T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 6.25T + 83T^{2} \)
89 \( 1 + 3.94T + 89T^{2} \)
97 \( 1 - 2.22T + 97T^{2} \)
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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

−8.185745431202137991593327972725, −6.87828739944025517587148011778, −6.68254947468065586189124643129, −5.91639562712300574859055368119, −5.08072794858601335465276047647, −4.70245388294689604740424542247, −3.76385319733403434319263237495, −3.09866433384340536361524567092, −1.51644407978302126561511551911, −0.45270432634263638809535743554, 0.45270432634263638809535743554, 1.51644407978302126561511551911, 3.09866433384340536361524567092, 3.76385319733403434319263237495, 4.70245388294689604740424542247, 5.08072794858601335465276047647, 5.91639562712300574859055368119, 6.68254947468065586189124643129, 6.87828739944025517587148011778, 8.185745431202137991593327972725

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