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  + 2.02·2-s − 2.93·3-s + 2.09·4-s − 5.94·6-s − 0.381·7-s + 0.196·8-s + 5.61·9-s + 0.280·11-s − 6.15·12-s + 4.00·13-s − 0.771·14-s − 3.79·16-s − 2.60·17-s + 11.3·18-s − 3.86·19-s + 1.11·21-s + 0.568·22-s − 0.698·23-s − 0.578·24-s + 8.10·26-s − 7.67·27-s − 0.799·28-s + 1.62·29-s + 9.73·31-s − 8.07·32-s − 0.824·33-s − 5.26·34-s + ⋯
L(s)  = 1  + 1.43·2-s − 1.69·3-s + 1.04·4-s − 2.42·6-s − 0.144·7-s + 0.0696·8-s + 1.87·9-s + 0.0846·11-s − 1.77·12-s + 1.11·13-s − 0.206·14-s − 0.948·16-s − 0.631·17-s + 2.67·18-s − 0.887·19-s + 0.244·21-s + 0.121·22-s − 0.145·23-s − 0.117·24-s + 1.58·26-s − 1.47·27-s − 0.151·28-s + 0.302·29-s + 1.74·31-s − 1.42·32-s − 0.143·33-s − 0.903·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.970226125$
$L(\frac12)$  $\approx$  $1.970226125$
$L(\frac{3}{2})$   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 \( 1 - 2.02T + 2T^{2} \)
3 \( 1 + 2.93T + 3T^{2} \)
7 \( 1 + 0.381T + 7T^{2} \)
11 \( 1 - 0.280T + 11T^{2} \)
13 \( 1 - 4.00T + 13T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 + 3.86T + 19T^{2} \)
23 \( 1 + 0.698T + 23T^{2} \)
29 \( 1 - 1.62T + 29T^{2} \)
31 \( 1 - 9.73T + 31T^{2} \)
37 \( 1 + 4.41T + 37T^{2} \)
41 \( 1 + 0.0157T + 41T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + 7.71T + 47T^{2} \)
53 \( 1 - 4.91T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 6.97T + 61T^{2} \)
67 \( 1 - 2.97T + 67T^{2} \)
71 \( 1 - 7.28T + 71T^{2} \)
73 \( 1 - 0.165T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 8.75T + 97T^{2} \)
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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

−7.83122907895880108911527586772, −6.63876451873421325418310711094, −6.42970535920460754863839059186, −5.97973488794424628885998920743, −5.13896647264478205450402227521, −4.56508019459693528864164982300, −4.04229937597475275732984103949, −3.09712916618321872082945148096, −1.90783121947290207155352741782, −0.64155995647407857965767449583, 0.64155995647407857965767449583, 1.90783121947290207155352741782, 3.09712916618321872082945148096, 4.04229937597475275732984103949, 4.56508019459693528864164982300, 5.13896647264478205450402227521, 5.97973488794424628885998920743, 6.42970535920460754863839059186, 6.63876451873421325418310711094, 7.83122907895880108911527586772

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