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Properties

Label 2-325-13.12-c1-0-10
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  − i·2-s + 2·3-s + 4-s − 2i·6-s + 5i·7-s − 3i·8-s + 9-s − 3i·11-s + 2·12-s + (3 + 2i)13-s + 5·14-s − 16-s − 5·17-s − i·18-s − 4i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.15·3-s + 0.5·4-s − 0.816i·6-s + 1.88i·7-s − 1.06i·8-s + 0.333·9-s − 0.904i·11-s + 0.577·12-s + (0.832 + 0.554i)13-s + 1.33·14-s − 0.250·16-s − 1.21·17-s − 0.235i·18-s − 0.917i·19-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

\(L(1)\) \(\approx\) \(1.99841 - 0.605070i\)
\(L(\frac12)\) \(\approx\) \(1.99841 - 0.605070i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 2T^{2} \)
3 \( 1 - 2T + 3T^{2} \)
7 \( 1 - 5iT - 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 - iT - 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 - 3iT - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 - 18iT - 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−11.46700607029575927390978402793, −10.87292610258284163388843425124, −9.242503401649232239745203316592, −9.007539828240496874666194551557, −8.093642427178320429863755108476, −6.60602589636245882480353366966, −5.70063770840894105669236331940, −3.84983222341498955142917010222, −2.71392006719634239188808586349, −2.10448523742543851980756786708, 1.88414572462430415752265888102, 3.42556616076850966040137692301, 4.45292740883216667540377214807, 6.14417059267584349737480582605, 7.12496940881709433883397922404, 7.81268210324789693045844564845, 8.483951197797741320423570850172, 9.851308563878474578219364909445, 10.61442414413858874068319526456, 11.53750914130824073006715677949

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