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Properties

Label 2-260-65.7-c1-0-3
Degree $2$
Conductor $260$
Sign $0.780 + 0.624i$
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  + (−0.0651 − 0.243i)3-s + (2.20 + 0.356i)5-s + (−2.48 − 4.30i)7-s + (2.54 − 1.46i)9-s + (1.07 + 4.01i)11-s + (−1.65 − 3.20i)13-s + (−0.0570 − 0.559i)15-s + (6.16 + 1.65i)17-s + (−1.01 − 0.270i)19-s + (−0.885 + 0.885i)21-s + (0.730 − 0.195i)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  + (−0.0375 − 0.140i)3-s + (0.987 + 0.159i)5-s + (−0.940 − 1.62i)7-s + (0.847 − 0.489i)9-s + (0.324 + 1.21i)11-s + (−0.459 − 0.888i)13-s + (−0.0147 − 0.144i)15-s + (1.49 + 0.400i)17-s + (−0.231 − 0.0621i)19-s + (−0.193 + 0.193i)21-s + (0.152 − 0.0408i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.780 + 0.624i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.780 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31703 - 0.461995i\)
\(L(\frac12)\) \(\approx\) \(1.31703 - 0.461995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 0.356i)T \)
13 \( 1 + (1.65 + 3.20i)T \)
good3 \( 1 + (0.0651 + 0.243i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.48 + 4.30i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.07 - 4.01i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-6.16 - 1.65i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.01 + 0.270i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.730 + 0.195i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (6.59 + 3.80i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.17 - 2.17i)T + 31iT^{2} \)
37 \( 1 + (3.09 - 5.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.57 - 0.689i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.779 - 2.90i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
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 \( 1 + (5.27 + 3.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.15 - 4.31i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 5.98iT - 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + 8.32T + 83T^{2} \)
89 \( 1 + (-1.99 + 0.534i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.23 - 4.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.18693926956158727460298360904, −10.39141123509828089698604583978, −10.11397991764302225235229806700, −9.462593921278218039113255053127, −7.57954636619280570942764825416, −7.00740521043856470399246249865, −6.00382083788265926854670139932, −4.49243477718915595647503232242, −3.28303906762169717239993122491, −1.32558776803533712616468055000, 1.99361398103570504685376016473, 3.34845453393445038396315583119, 5.21394386498351921251547236597, 5.87818361050992215675633667959, 6.93218912546156668789235959718, 8.495504508580875618789539881146, 9.414740499276343767563364997992, 9.843607573264127283069750216264, 11.16944252551108958772386460709, 12.27635508753140577503915130387

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