Properties

Label 2-325-1.1-c1-0-13
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  + 2.67·2-s − 0.481·3-s + 5.15·4-s − 1.28·6-s + 0.806·7-s + 8.44·8-s − 2.76·9-s − 3.67·11-s − 2.48·12-s − 13-s + 2.15·14-s + 12.2·16-s − 1.35·17-s − 7.40·18-s − 1.67·19-s − 0.387·21-s − 9.83·22-s + 6.48·23-s − 4.06·24-s − 2.67·26-s + 2.77·27-s + 4.15·28-s + 2.41·29-s − 5.28·31-s + 15.9·32-s + 1.76·33-s − 3.61·34-s + ⋯
L(s)  = 1  + 1.89·2-s − 0.277·3-s + 2.57·4-s − 0.525·6-s + 0.304·7-s + 2.98·8-s − 0.922·9-s − 1.10·11-s − 0.716·12-s − 0.277·13-s + 0.576·14-s + 3.06·16-s − 0.327·17-s − 1.74·18-s − 0.384·19-s − 0.0846·21-s − 2.09·22-s + 1.35·23-s − 0.829·24-s − 0.524·26-s + 0.534·27-s + 0.785·28-s + 0.449·29-s − 0.949·31-s + 2.81·32-s + 0.307·33-s − 0.619·34-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

\(L(1)\) \(\approx\) \(3.324006728\)
\(L(\frac12)\) \(\approx\) \(3.324006728\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.67T + 2T^{2} \)
3 \( 1 + 0.481T + 3T^{2} \)
7 \( 1 - 0.806T + 7T^{2} \)
11 \( 1 + 3.67T + 11T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 + 1.67T + 19T^{2} \)
23 \( 1 - 6.48T + 23T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 - 3.76T + 37T^{2} \)
41 \( 1 + 8.31T + 41T^{2} \)
43 \( 1 + 6.79T + 43T^{2} \)
47 \( 1 - 3.19T + 47T^{2} \)
53 \( 1 - 5.73T + 53T^{2} \)
59 \( 1 - 5.98T + 59T^{2} \)
61 \( 1 + 1.76T + 61T^{2} \)
67 \( 1 - 9.89T + 67T^{2} \)
71 \( 1 - 8.56T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 2.26T + 79T^{2} \)
83 \( 1 - 3.84T + 83T^{2} \)
89 \( 1 - 2.77T + 89T^{2} \)
97 \( 1 + 1.87T + 97T^{2} \)
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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

−11.69038210791261118237189109974, −11.14388329477918097583508211310, −10.31595839357296161585415305426, −8.535963932891947942564167190392, −7.36695212403749232505614030953, −6.38944427101187592256209365279, −5.33062873663749177208229155137, −4.82844588955254187264925613732, −3.35571709981399726265261374113, −2.33461534776682996841476069623, 2.33461534776682996841476069623, 3.35571709981399726265261374113, 4.82844588955254187264925613732, 5.33062873663749177208229155137, 6.38944427101187592256209365279, 7.36695212403749232505614030953, 8.535963932891947942564167190392, 10.31595839357296161585415305426, 11.14388329477918097583508211310, 11.69038210791261118237189109974

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