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Properties

Label 2-260-65.37-c1-0-3
Degree $2$
Conductor $260$
Sign $0.836 + 0.547i$
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  + (−0.5 − 0.133i)3-s + (2 − i)5-s + (0.133 − 0.232i)7-s + (−2.36 − 1.36i)9-s + (4.59 + 1.23i)11-s + (2 − 3i)13-s + (−1.13 + 0.232i)15-s + (−0.767 − 2.86i)17-s + (0.866 + 3.23i)19-s + (−0.0980 + 0.0980i)21-s + (−0.0358 + 0.133i)23-s + (3 − 4i)25-s + (2.09 + 2.09i)27-s + (1.03 − 0.598i)29-s + (2.26 + 2.26i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.0773i)3-s + (0.894 − 0.447i)5-s + (0.0506 − 0.0877i)7-s + (−0.788 − 0.455i)9-s + (1.38 + 0.371i)11-s + (0.554 − 0.832i)13-s + (−0.292 + 0.0599i)15-s + (−0.186 − 0.695i)17-s + (0.198 + 0.741i)19-s + (−0.0214 + 0.0214i)21-s + (−0.00748 + 0.0279i)23-s + (0.600 − 0.800i)25-s + (0.403 + 0.403i)27-s + (0.192 − 0.111i)29-s + (0.407 + 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.836 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28023 - 0.381594i\)
\(L(\frac12)\) \(\approx\) \(1.28023 - 0.381594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
13 \( 1 + (-2 + 3i)T \)
good3 \( 1 + (0.5 + 0.133i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.133 + 0.232i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.59 - 1.23i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.767 + 2.86i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.866 - 3.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.0358 - 0.133i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.03 + 0.598i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.26 - 2.26i)T + 31iT^{2} \)
37 \( 1 + (1.86 + 3.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.86 - 6.96i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.96 - 2.66i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 7.46T + 47T^{2} \)
53 \( 1 + (8.46 - 8.46i)T - 53iT^{2} \)
59 \( 1 + (13.7 - 3.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.7 + 6.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.86 + 0.767i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 0.928iT - 73T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (-0.794 + 2.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.13 - 1.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.97120854600435356301947820279, −11.03539383662011132687699117274, −9.852836593267286868945611235589, −9.134575911685515433290605654606, −8.161351086868398573077112588103, −6.61005388400809259489562530219, −5.93835220952622605392422577653, −4.77956201674702098152430519378, −3.21055211573303064445097567141, −1.33051755301066278508759502655, 1.85028993952903655405457105834, 3.45780390356201486857207431742, 4.99041875014808462142456253639, 6.20638442546027418265760043616, 6.72453033726633660505346731879, 8.444660537791265423144972916360, 9.163823511361002364910823398192, 10.23948139872712001142301398732, 11.26140567765168359543531537528, 11.74113245501623722643074906252

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