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  − 2.01·2-s − 0.500·3-s + 2.04·4-s + 1.00·6-s + 0.852·7-s − 0.0822·8-s − 2.74·9-s + 0.719·11-s − 1.02·12-s + 1.93·13-s − 1.71·14-s − 3.91·16-s − 0.439·17-s + 5.52·18-s + 5.85·19-s − 0.426·21-s − 1.44·22-s + 7.09·23-s + 0.0411·24-s − 3.88·26-s + 2.87·27-s + 1.73·28-s + 10.0·29-s + 5.69·31-s + 8.03·32-s − 0.360·33-s + 0.884·34-s + ⋯
L(s)  = 1  − 1.42·2-s − 0.289·3-s + 1.02·4-s + 0.410·6-s + 0.322·7-s − 0.0290·8-s − 0.916·9-s + 0.216·11-s − 0.294·12-s + 0.536·13-s − 0.457·14-s − 0.979·16-s − 0.106·17-s + 1.30·18-s + 1.34·19-s − 0.0931·21-s − 0.308·22-s + 1.47·23-s + 0.00840·24-s − 0.762·26-s + 0.553·27-s + 0.328·28-s + 1.86·29-s + 1.02·31-s + 1.42·32-s − 0.0627·33-s + 0.151·34-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  =  $(2,\ 6025,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9131249016$
$L(\frac12)$  $\approx$  $0.9131249016$
$L(\frac{3}{2})$   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 \( 1 + 2.01T + 2T^{2} \)
3 \( 1 + 0.500T + 3T^{2} \)
7 \( 1 - 0.852T + 7T^{2} \)
11 \( 1 - 0.719T + 11T^{2} \)
13 \( 1 - 1.93T + 13T^{2} \)
17 \( 1 + 0.439T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 - 7.09T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 - 5.69T + 31T^{2} \)
37 \( 1 - 3.17T + 37T^{2} \)
41 \( 1 + 6.39T + 41T^{2} \)
43 \( 1 + 4.29T + 43T^{2} \)
47 \( 1 - 0.642T + 47T^{2} \)
53 \( 1 + 0.729T + 53T^{2} \)
59 \( 1 - 0.348T + 59T^{2} \)
61 \( 1 - 1.12T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 - 0.552T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 6.62T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 5.00T + 97T^{2} \)
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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

−8.240601074802155763784936904333, −7.62679565733569647478372939815, −6.72383516548502431564428342791, −6.29448778701371256801919167297, −5.14437044898968999905235777540, −4.70775796479903337283086190688, −3.34049531657156864814397908981, −2.62846812081580662832380849379, −1.36163713423395386192194053971, −0.71756481672008946420241088002, 0.71756481672008946420241088002, 1.36163713423395386192194053971, 2.62846812081580662832380849379, 3.34049531657156864814397908981, 4.70775796479903337283086190688, 5.14437044898968999905235777540, 6.29448778701371256801919167297, 6.72383516548502431564428342791, 7.62679565733569647478372939815, 8.240601074802155763784936904333

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