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Properties

Label 2-260-65.64-c1-0-1
Degree $2$
Conductor $260$
Sign $0.138 - 0.990i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
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  + 0.646i·3-s + (−1.73 + 1.41i)5-s + 0.913·7-s + 2.58·9-s + 3.94i·11-s + (−2.64 + 2.44i)13-s + (−0.913 − 1.11i)15-s + 6.19i·17-s + 1.11i·19-s + 0.590i·21-s − 5.54i·23-s + (0.999 − 4.89i)25-s + 3.60i·27-s + 1.58·29-s − 9.60i·31-s + ⋯
L(s)  = 1  + 0.373i·3-s + (−0.774 + 0.632i)5-s + 0.345·7-s + 0.860·9-s + 1.19i·11-s + (−0.733 + 0.679i)13-s + (−0.235 − 0.288i)15-s + 1.50i·17-s + 0.256i·19-s + 0.128i·21-s − 1.15i·23-s + (0.199 − 0.979i)25-s + 0.694i·27-s + 0.293·29-s − 1.72i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845394 + 0.735222i\)
\(L(\frac12)\) \(\approx\) \(0.845394 + 0.735222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.73 - 1.41i)T \)
13 \( 1 + (2.64 - 2.44i)T \)
good3 \( 1 - 0.646iT - 3T^{2} \)
7 \( 1 - 0.913T + 7T^{2} \)
11 \( 1 - 3.94iT - 11T^{2} \)
17 \( 1 - 6.19iT - 17T^{2} \)
19 \( 1 - 1.11iT - 19T^{2} \)
23 \( 1 + 5.54iT - 23T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 + 9.60iT - 31T^{2} \)
37 \( 1 - 7.84T + 37T^{2} \)
41 \( 1 - 5.06iT - 41T^{2} \)
43 \( 1 + 6.83iT - 43T^{2} \)
47 \( 1 - 9.66T + 47T^{2} \)
53 \( 1 - 1.29iT - 53T^{2} \)
59 \( 1 + 9.01iT - 59T^{2} \)
61 \( 1 + 5.58T + 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + 6.77iT - 71T^{2} \)
73 \( 1 - 7.84T + 73T^{2} \)
79 \( 1 + 7.16T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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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

−12.24666806844342382894836474551, −11.17383760691284090492114957852, −10.29603055098451163354038259788, −9.576425159962685039096404467491, −8.148136371961063424654358351029, −7.33607778601864873893690617050, −6.38075984860263469939575140732, −4.57132868850683820827067915798, −4.03780666551540051716002148793, −2.17168941959692474062811147196, 0.956121926463777211496794486991, 3.07030047949319203839179817174, 4.52780398632788205743878440926, 5.49405522674696637756099548560, 7.10291956217587759723370862750, 7.75548715300287642027422543012, 8.765043914982743935466726511438, 9.790740068878292673804146077642, 11.05563889804344644746898681765, 11.81627476328985387285982128698

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