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  + 1.32·2-s + 2.18·3-s − 0.231·4-s + 2.90·6-s + 3.83·7-s − 2.96·8-s + 1.75·9-s + 4.78·11-s − 0.504·12-s − 1.75·13-s + 5.09·14-s − 3.48·16-s + 6.01·17-s + 2.33·18-s − 4.20·19-s + 8.36·21-s + 6.35·22-s − 4.28·23-s − 6.47·24-s − 2.33·26-s − 2.70·27-s − 0.887·28-s + 4.96·29-s + 4.59·31-s + 1.30·32-s + 10.4·33-s + 8.00·34-s + ⋯
L(s)  = 1  + 0.940·2-s + 1.25·3-s − 0.115·4-s + 1.18·6-s + 1.44·7-s − 1.04·8-s + 0.586·9-s + 1.44·11-s − 0.145·12-s − 0.487·13-s + 1.36·14-s − 0.870·16-s + 1.46·17-s + 0.551·18-s − 0.963·19-s + 1.82·21-s + 1.35·22-s − 0.893·23-s − 1.32·24-s − 0.458·26-s − 0.521·27-s − 0.167·28-s + 0.921·29-s + 0.825·31-s + 0.230·32-s + 1.81·33-s + 1.37·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.757946643$
$L(\frac12)$  $\approx$  $5.757946643$
$L(\frac{3}{2})$   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 \( 1 - 1.32T + 2T^{2} \)
3 \( 1 - 2.18T + 3T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 - 4.78T + 11T^{2} \)
13 \( 1 + 1.75T + 13T^{2} \)
17 \( 1 - 6.01T + 17T^{2} \)
19 \( 1 + 4.20T + 19T^{2} \)
23 \( 1 + 4.28T + 23T^{2} \)
29 \( 1 - 4.96T + 29T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 - 9.09T + 43T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 - 1.36T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 9.85T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 0.162T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 5.77T + 83T^{2} \)
89 \( 1 - 4.73T + 89T^{2} \)
97 \( 1 + 0.439T + 97T^{2} \)
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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

−8.024938029969711847453445560205, −7.71255830438646972631711306492, −6.45495745645658270798379504698, −5.91139103900369427626898181572, −4.88042357387449298658201341338, −4.35255109677486021074832805153, −3.78323010642773772292521263261, −2.92985458169466542003687784958, −2.13019004690286069492745712142, −1.12282353191083303328075216364, 1.12282353191083303328075216364, 2.13019004690286069492745712142, 2.92985458169466542003687784958, 3.78323010642773772292521263261, 4.35255109677486021074832805153, 4.88042357387449298658201341338, 5.91139103900369427626898181572, 6.45495745645658270798379504698, 7.71255830438646972631711306492, 8.024938029969711847453445560205

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