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Properties

Label 2-260-65.33-c1-0-1
Degree $2$
Conductor $260$
Sign $-0.109 - 0.994i$
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  + (0.243 − 0.0651i)3-s + (0.356 + 2.20i)5-s + (−4.30 + 2.48i)7-s + (−2.54 + 1.46i)9-s + (1.07 + 4.01i)11-s + (3.20 − 1.65i)13-s + (0.230 + 0.513i)15-s + (1.65 − 6.16i)17-s + (1.01 + 0.270i)19-s + (−0.885 + 0.885i)21-s + (0.195 + 0.730i)23-s + (−4.74 + 1.57i)25-s + (−1.05 + 1.05i)27-s + (6.59 + 3.80i)29-s + (2.17 + 2.17i)31-s + ⋯
L(s)  = 1  + (0.140 − 0.0375i)3-s + (0.159 + 0.987i)5-s + (−1.62 + 0.940i)7-s + (−0.847 + 0.489i)9-s + (0.324 + 1.21i)11-s + (0.888 − 0.459i)13-s + (0.0594 + 0.132i)15-s + (0.400 − 1.49i)17-s + (0.231 + 0.0621i)19-s + (−0.193 + 0.193i)21-s + (0.0408 + 0.152i)23-s + (−0.949 + 0.314i)25-s + (−0.203 + 0.203i)27-s + (1.22 + 0.706i)29-s + (0.390 + 0.390i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.109 - 0.994i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.109 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681873 + 0.760791i\)
\(L(\frac12)\) \(\approx\) \(0.681873 + 0.760791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.356 - 2.20i)T \)
13 \( 1 + (-3.20 + 1.65i)T \)
good3 \( 1 + (-0.243 + 0.0651i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (4.30 - 2.48i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.07 - 4.01i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.65 + 6.16i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.01 - 0.270i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.195 - 0.730i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-6.59 - 3.80i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.17 - 2.17i)T + 31iT^{2} \)
37 \( 1 + (-5.36 - 3.09i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.57 - 0.689i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.90 + 0.779i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 1.62iT - 47T^{2} \)
53 \( 1 + (5.32 + 5.32i)T + 53iT^{2} \)
59 \( 1 + (-2.52 + 9.43i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.39 - 9.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.04 - 5.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.15 - 4.31i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 5.98T + 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 + 8.32iT - 83T^{2} \)
89 \( 1 + (1.99 - 0.534i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.17 - 7.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.17614362880007508657163141276, −11.39608422686757213911566796801, −10.10741734472681114414236586166, −9.579574071770131106338299689101, −8.461943473617902563278296611590, −7.08852909058562221221086818038, −6.35052635060878790420697593036, −5.28283128841890668683678532959, −3.29002455466659359925234096891, −2.61086426622704848495500029336, 0.799381073872753169740330827746, 3.27460503764728039820001165498, 4.08602240146916059361175829464, 6.05167127888850021965667272658, 6.29010239830171365546968031218, 8.090837725375692807618764593413, 8.874427877635879155317485681321, 9.696126572864970307443332021679, 10.71656845035548446747980820279, 11.83542329905021458427384802487

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