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  + 2.59·2-s + 1.20·3-s + 4.74·4-s + 3.13·6-s + 0.744·7-s + 7.12·8-s − 1.54·9-s + 6.28·11-s + 5.71·12-s + 4.06·13-s + 1.93·14-s + 9.02·16-s + 1.60·17-s − 4.01·18-s + 2.14·19-s + 0.897·21-s + 16.3·22-s − 9.25·23-s + 8.59·24-s + 10.5·26-s − 5.48·27-s + 3.53·28-s + 3.13·29-s − 3.15·31-s + 9.17·32-s + 7.57·33-s + 4.15·34-s + ⋯
L(s)  = 1  + 1.83·2-s + 0.695·3-s + 2.37·4-s + 1.27·6-s + 0.281·7-s + 2.52·8-s − 0.515·9-s + 1.89·11-s + 1.65·12-s + 1.12·13-s + 0.516·14-s + 2.25·16-s + 0.388·17-s − 0.947·18-s + 0.491·19-s + 0.195·21-s + 3.48·22-s − 1.92·23-s + 1.75·24-s + 2.07·26-s − 1.05·27-s + 0.667·28-s + 0.581·29-s − 0.566·31-s + 1.62·32-s + 1.31·33-s + 0.712·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.797679731$
$L(\frac12)$  $\approx$  $9.797679731$
$L(\frac{3}{2})$   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 \( 1 - 2.59T + 2T^{2} \)
3 \( 1 - 1.20T + 3T^{2} \)
7 \( 1 - 0.744T + 7T^{2} \)
11 \( 1 - 6.28T + 11T^{2} \)
13 \( 1 - 4.06T + 13T^{2} \)
17 \( 1 - 1.60T + 17T^{2} \)
19 \( 1 - 2.14T + 19T^{2} \)
23 \( 1 + 9.25T + 23T^{2} \)
29 \( 1 - 3.13T + 29T^{2} \)
31 \( 1 + 3.15T + 31T^{2} \)
37 \( 1 + 4.19T + 37T^{2} \)
41 \( 1 + 4.52T + 41T^{2} \)
43 \( 1 + 6.99T + 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 + 3.71T + 53T^{2} \)
59 \( 1 - 3.34T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 5.80T + 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 1.64T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 4.74T + 83T^{2} \)
89 \( 1 + 7.95T + 89T^{2} \)
97 \( 1 - 5.49T + 97T^{2} \)
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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

−8.053154266233791052086863439525, −7.04175548207397866359502796380, −6.41672858328065585629105686586, −5.89381214428857316065976240264, −5.18477606063700190823359333910, −4.15550603325916963238695765866, −3.68795883511494264094841482174, −3.25216199391103844217759163150, −2.09547666587401544829821476218, −1.43589432747494202326951144168, 1.43589432747494202326951144168, 2.09547666587401544829821476218, 3.25216199391103844217759163150, 3.68795883511494264094841482174, 4.15550603325916963238695765866, 5.18477606063700190823359333910, 5.89381214428857316065976240264, 6.41672858328065585629105686586, 7.04175548207397866359502796380, 8.053154266233791052086863439525

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