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Properties

Label 2-260-65.2-c1-0-1
Degree $2$
Conductor $260$
Sign $0.743 - 0.669i$
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  + (2.49 + 0.668i)3-s + (−0.380 + 2.20i)5-s + (−0.749 − 0.432i)7-s + (3.17 + 1.83i)9-s + (0.417 − 1.55i)11-s + (1.32 + 3.35i)13-s + (−2.42 + 5.23i)15-s + (−1.33 − 4.99i)17-s + (−1.16 + 0.312i)19-s + (−1.57 − 1.57i)21-s + (0.704 − 2.62i)23-s + (−4.70 − 1.67i)25-s + (1.20 + 1.20i)27-s + (5.94 − 3.43i)29-s + (0.191 − 0.191i)31-s + ⋯
L(s)  = 1  + (1.43 + 0.385i)3-s + (−0.170 + 0.985i)5-s + (−0.283 − 0.163i)7-s + (1.05 + 0.610i)9-s + (0.125 − 0.469i)11-s + (0.367 + 0.930i)13-s + (−0.625 + 1.35i)15-s + (−0.324 − 1.21i)17-s + (−0.267 + 0.0716i)19-s + (−0.344 − 0.344i)21-s + (0.146 − 0.548i)23-s + (−0.941 − 0.335i)25-s + (0.232 + 0.232i)27-s + (1.10 − 0.636i)29-s + (0.0344 − 0.0344i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.743 - 0.669i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.743 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70878 + 0.655907i\)
\(L(\frac12)\) \(\approx\) \(1.70878 + 0.655907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.380 - 2.20i)T \)
13 \( 1 + (-1.32 - 3.35i)T \)
good3 \( 1 + (-2.49 - 0.668i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.749 + 0.432i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.417 + 1.55i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.33 + 4.99i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.16 - 0.312i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.704 + 2.62i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.94 + 3.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.191 + 0.191i)T - 31iT^{2} \)
37 \( 1 + (8.22 - 4.74i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.417 - 0.111i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-11.5 + 3.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.52iT - 47T^{2} \)
53 \( 1 + (4.88 - 4.88i)T - 53iT^{2} \)
59 \( 1 + (3.05 + 11.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.94 - 5.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.76 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.85 + 14.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 3.98iT - 79T^{2} \)
83 \( 1 - 6.25iT - 83T^{2} \)
89 \( 1 + (9.80 + 2.62i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.728 - 1.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.00580301416414053825731269493, −11.01374008764399445253038527837, −10.04477621462085687353983605639, −9.174745287370152793839208906856, −8.360982849554883225398917753544, −7.25708172725296189080278172937, −6.36230593807193285365466305795, −4.41456264279390877459061441710, −3.36416117699502050331695481226, −2.43813812939997688294197243417, 1.64049670387828563455720561482, 3.14265518710272768852270895888, 4.30722795088162552845075209150, 5.79824289146974833287861697439, 7.25120894909253776742273574460, 8.242046499733314054020234700119, 8.754688457928192485347170743586, 9.645177157165021448296640422111, 10.81869172408251662761682573942, 12.43616992725517350730367288426

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