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  − 0.115·2-s + 3.28·3-s − 1.98·4-s − 0.379·6-s − 3.19·7-s + 0.461·8-s + 7.77·9-s + 1.38·11-s − 6.52·12-s + 5.87·13-s + 0.369·14-s + 3.91·16-s − 5.28·17-s − 0.899·18-s + 4.99·19-s − 10.4·21-s − 0.160·22-s − 3.07·23-s + 1.51·24-s − 0.679·26-s + 15.6·27-s + 6.35·28-s + 3.28·29-s + 0.672·31-s − 1.37·32-s + 4.56·33-s + 0.611·34-s + ⋯
L(s)  = 1  − 0.0817·2-s + 1.89·3-s − 0.993·4-s − 0.155·6-s − 1.20·7-s + 0.163·8-s + 2.59·9-s + 0.419·11-s − 1.88·12-s + 1.62·13-s + 0.0988·14-s + 0.979·16-s − 1.28·17-s − 0.212·18-s + 1.14·19-s − 2.28·21-s − 0.0342·22-s − 0.640·23-s + 0.309·24-s − 0.133·26-s + 3.01·27-s + 1.20·28-s + 0.610·29-s + 0.120·31-s − 0.243·32-s + 0.794·33-s + 0.104·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.122178924$
$L(\frac12)$  $\approx$  $3.122178924$
$L(\frac{3}{2})$   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 \( 1 + 0.115T + 2T^{2} \)
3 \( 1 - 3.28T + 3T^{2} \)
7 \( 1 + 3.19T + 7T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 - 5.87T + 13T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 - 4.99T + 19T^{2} \)
23 \( 1 + 3.07T + 23T^{2} \)
29 \( 1 - 3.28T + 29T^{2} \)
31 \( 1 - 0.672T + 31T^{2} \)
37 \( 1 - 3.79T + 37T^{2} \)
41 \( 1 - 0.970T + 41T^{2} \)
43 \( 1 + 7.93T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 8.45T + 53T^{2} \)
59 \( 1 - 5.70T + 59T^{2} \)
61 \( 1 - 0.717T + 61T^{2} \)
67 \( 1 + 8.81T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 8.75T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 1.18T + 97T^{2} \)
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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

−8.306786298371722780284242594766, −7.68199274647943811927062716337, −6.67556718241314105543048148071, −6.23241649796549713590895903585, −4.90958320987750029423872147565, −4.07521071723478479893693598837, −3.53235551557354699969678916681, −3.10178665422785639102991551509, −1.94761472284328676870295021048, −0.898275218291623688770780709962, 0.898275218291623688770780709962, 1.94761472284328676870295021048, 3.10178665422785639102991551509, 3.53235551557354699969678916681, 4.07521071723478479893693598837, 4.90958320987750029423872147565, 6.23241649796549713590895903585, 6.67556718241314105543048148071, 7.68199274647943811927062716337, 8.306786298371722780284242594766

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