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  + 1.28·2-s + 0.126·3-s − 0.345·4-s + 0.162·6-s − 1.03·7-s − 3.01·8-s − 2.98·9-s + 0.227·11-s − 0.0435·12-s − 3.38·13-s − 1.32·14-s − 3.18·16-s − 7.12·17-s − 3.83·18-s + 3.40·19-s − 0.130·21-s + 0.293·22-s − 6.91·23-s − 0.380·24-s − 4.34·26-s − 0.755·27-s + 0.356·28-s + 0.569·29-s + 4.93·31-s + 1.93·32-s + 0.0287·33-s − 9.16·34-s + ⋯
L(s)  = 1  + 0.909·2-s + 0.0728·3-s − 0.172·4-s + 0.0662·6-s − 0.389·7-s − 1.06·8-s − 0.994·9-s + 0.0687·11-s − 0.0125·12-s − 0.937·13-s − 0.354·14-s − 0.797·16-s − 1.72·17-s − 0.904·18-s + 0.780·19-s − 0.0284·21-s + 0.0625·22-s − 1.44·23-s − 0.0777·24-s − 0.852·26-s − 0.145·27-s + 0.0672·28-s + 0.105·29-s + 0.886·31-s + 0.341·32-s + 0.00500·33-s − 1.57·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.378936551$
$L(\frac12)$  $\approx$  $1.378936551$
$L(\frac{3}{2})$   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 \( 1 - 1.28T + 2T^{2} \)
3 \( 1 - 0.126T + 3T^{2} \)
7 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 - 0.227T + 11T^{2} \)
13 \( 1 + 3.38T + 13T^{2} \)
17 \( 1 + 7.12T + 17T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 + 6.91T + 23T^{2} \)
29 \( 1 - 0.569T + 29T^{2} \)
31 \( 1 - 4.93T + 31T^{2} \)
37 \( 1 - 5.37T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 0.910T + 43T^{2} \)
47 \( 1 - 8.50T + 47T^{2} \)
53 \( 1 - 7.76T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 3.65T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 9.48T + 71T^{2} \)
73 \( 1 - 7.10T + 73T^{2} \)
79 \( 1 - 0.366T + 79T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + 7.54T + 89T^{2} \)
97 \( 1 - 7.85T + 97T^{2} \)
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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

−8.143428440481123408350879489663, −7.24721841762354553556253936294, −6.38565095051655275357303245564, −5.87949695375732592987709601796, −5.17843821974442518388008635343, −4.35727901118106054646438336033, −3.86500991176374422709242163041, −2.70288945399940508326418172493, −2.42609835019884216306503343455, −0.50076888313895986608905428779, 0.50076888313895986608905428779, 2.42609835019884216306503343455, 2.70288945399940508326418172493, 3.86500991176374422709242163041, 4.35727901118106054646438336033, 5.17843821974442518388008635343, 5.87949695375732592987709601796, 6.38565095051655275357303245564, 7.24721841762354553556253936294, 8.143428440481123408350879489663

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