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Properties

Label 2-260-260.243-c1-0-21
Degree $2$
Conductor $260$
Sign $0.196 - 0.980i$
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  + (1.31 + 0.524i)2-s + (0.689 + 2.57i)3-s + (1.45 + 1.37i)4-s + (0.222 − 2.22i)5-s + (−0.443 + 3.73i)6-s + (0.280 − 1.04i)7-s + (1.18 + 2.56i)8-s + (−3.54 + 2.04i)9-s + (1.45 − 2.80i)10-s + (0.213 + 0.123i)11-s + (−2.54 + 4.67i)12-s + (−1.52 − 3.26i)13-s + (0.915 − 1.22i)14-s + (5.87 − 0.960i)15-s + (0.206 + 3.99i)16-s + (−5.82 − 1.56i)17-s + ⋯
L(s)  = 1  + (0.928 + 0.370i)2-s + (0.397 + 1.48i)3-s + (0.725 + 0.688i)4-s + (0.0995 − 0.995i)5-s + (−0.180 + 1.52i)6-s + (0.105 − 0.395i)7-s + (0.418 + 0.908i)8-s + (−1.18 + 0.681i)9-s + (0.461 − 0.887i)10-s + (0.0643 + 0.0371i)11-s + (−0.733 + 1.35i)12-s + (−0.422 − 0.906i)13-s + (0.244 − 0.327i)14-s + (1.51 − 0.248i)15-s + (0.0516 + 0.998i)16-s + (−1.41 − 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.196 - 0.980i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79868 + 1.47458i\)
\(L(\frac12)\) \(\approx\) \(1.79868 + 1.47458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 0.524i)T \)
5 \( 1 + (-0.222 + 2.22i)T \)
13 \( 1 + (1.52 + 3.26i)T \)
good3 \( 1 + (-0.689 - 2.57i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.280 + 1.04i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.213 - 0.123i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.82 + 1.56i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.97 + 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.742 - 2.77i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.76 - 2.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.881iT - 31T^{2} \)
37 \( 1 + (-1.32 - 4.95i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.89 + 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.93 + 0.517i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.04 - 6.04i)T + 47iT^{2} \)
53 \( 1 + (0.436 + 0.436i)T + 53iT^{2} \)
59 \( 1 + (-5.04 - 8.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.71 + 4.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.5 - 3.36i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.10 + 4.68i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.41 + 3.41i)T + 73iT^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + (10.8 - 10.8i)T - 83iT^{2} \)
89 \( 1 + (1.31 + 0.762i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.4 - 3.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.36323127199045916358652227858, −11.18884061899856611342899530779, −10.39970032642396773821947342594, −9.195663574378974571912463262430, −8.494261554627783079401953856922, −7.23437342470550425865376130717, −5.66845879817162848988811337945, −4.69681427528726139594112514108, −4.18359063411986620716286398733, −2.75291976972207859605289931095, 1.94036636672734558034607151595, 2.65803903150085082939188305653, 4.25606283741406335054432390579, 6.05871060801715433420783354410, 6.63828850559671813085270046441, 7.45396672616064959706144677602, 8.735376967093141991954459310108, 10.15791089957909205962505671828, 11.24284925781545636058987834309, 11.93919493156843438097760046651

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