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Properties

Label 2-260-65.28-c1-0-2
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: $\chi_{260} (93, \cdot )$
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

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

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