Properties

Label 2-325-1.1-c1-0-1
Degree $2$
Conductor $325$
Sign $1$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·2-s − 2.82·3-s − 1.82·4-s − 1.17·6-s + 0.414·7-s − 1.58·8-s + 5.00·9-s + 3.58·11-s + 5.17·12-s − 13-s + 0.171·14-s + 3·16-s − 1.82·17-s + 2.07·18-s + 7.65·19-s − 1.17·21-s + 1.48·22-s + 8.82·23-s + 4.48·24-s − 0.414·26-s − 5.65·27-s − 0.757·28-s − 5.82·29-s − 7.24·31-s + 4.41·32-s − 10.1·33-s − 0.757·34-s + ⋯
L(s)  = 1  + 0.292·2-s − 1.63·3-s − 0.914·4-s − 0.478·6-s + 0.156·7-s − 0.560·8-s + 1.66·9-s + 1.08·11-s + 1.49·12-s − 0.277·13-s + 0.0458·14-s + 0.750·16-s − 0.443·17-s + 0.488·18-s + 1.75·19-s − 0.255·21-s + 0.316·22-s + 1.84·23-s + 0.915·24-s − 0.0812·26-s − 1.08·27-s − 0.143·28-s − 1.08·29-s − 1.30·31-s + 0.780·32-s − 1.76·33-s − 0.129·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7425311965\)
\(L(\frac12)\) \(\approx\) \(0.7425311965\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + T \)
good2 \( 1 - 0.414T + 2T^{2} \)
3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 - 0.414T + 7T^{2} \)
11 \( 1 - 3.58T + 11T^{2} \)
17 \( 1 + 1.82T + 17T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
29 \( 1 + 5.82T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 5.65T + 41T^{2} \)
43 \( 1 - 4.82T + 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 4.75T + 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 8.48T + 89T^{2} \)
97 \( 1 + 4.82T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.45255025936690215557238566624, −11.11477908981489042514208322709, −9.661693432332869129548718238911, −9.169947437687977848265208195093, −7.53607462685484137032843166972, −6.51374032857296880535367423457, −5.44217524763044622936212804396, −4.86658875375843851726464671663, −3.66338328472537411261628031991, −0.951268133424611537112902673213, 0.951268133424611537112902673213, 3.66338328472537411261628031991, 4.86658875375843851726464671663, 5.44217524763044622936212804396, 6.51374032857296880535367423457, 7.53607462685484137032843166972, 9.169947437687977848265208195093, 9.661693432332869129548718238911, 11.11477908981489042514208322709, 11.45255025936690215557238566624

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