Properties

Label 2-325-13.12-c1-0-1
Degree $2$
Conductor $325$
Sign $-0.832 + 0.554i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 3-s − 2·4-s − 2i·6-s + 2i·7-s − 2·9-s + 2·12-s + (−3 + 2i)13-s − 4·14-s − 4·16-s − 2·17-s − 4i·18-s − 4i·19-s − 2i·21-s − 23-s + ⋯
L(s)  = 1  + 1.41i·2-s − 0.577·3-s − 4-s − 0.816i·6-s + 0.755i·7-s − 0.666·9-s + 0.577·12-s + (−0.832 + 0.554i)13-s − 1.06·14-s − 16-s − 0.485·17-s − 0.942i·18-s − 0.917i·19-s − 0.436i·21-s − 0.208·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191891 - 0.633773i\)
\(L(\frac12)\) \(\approx\) \(0.191891 - 0.633773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 2i)T \)
good2 \( 1 - 2iT - 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−11.93176266694298933058460370640, −11.51797952032722027855627646741, −10.19355003038935935112000501799, −8.867745395695592108036724025108, −8.442749595770913852285471177392, −7.02886080438866008384258629381, −6.47250326800870221202037188489, −5.37167074814700313175886168848, −4.76194707141289287292046691991, −2.61406413337654857103539023048, 0.48137645504867270155404486414, 2.27880256373408315256147343358, 3.56889414762037624882687025841, 4.68951004590417022965209666797, 5.96843779068964231574949881864, 7.20569607189996205962244713118, 8.426577148881573092317124951724, 9.703465973113253747598113452150, 10.36184021706980554193114109300, 11.08018920854494500666756217377

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