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Properties

Label 2-260-65.47-c1-0-5
Degree $2$
Conductor $260$
Sign $0.749 + 0.661i$
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  + (1.61 − 1.61i)3-s + 2.23·5-s − 2.23i·9-s + (−0.381 + 0.381i)11-s + (−3 + 2i)13-s + (3.61 − 3.61i)15-s + (−2.23 + 2.23i)17-s + (0.854 − 0.854i)19-s + (−5.61 − 5.61i)23-s + 5.00·25-s + (1.23 + 1.23i)27-s − 0.763i·29-s + (−0.854 − 0.854i)31-s + 1.23i·33-s − 3.70·37-s + ⋯
L(s)  = 1  + (0.934 − 0.934i)3-s + 0.999·5-s − 0.745i·9-s + (−0.115 + 0.115i)11-s + (−0.832 + 0.554i)13-s + (0.934 − 0.934i)15-s + (−0.542 + 0.542i)17-s + (0.195 − 0.195i)19-s + (−1.17 − 1.17i)23-s + 1.00·25-s + (0.237 + 0.237i)27-s − 0.141i·29-s + (−0.153 − 0.153i)31-s + 0.215i·33-s − 0.609·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70251 - 0.643964i\)
\(L(\frac12)\) \(\approx\) \(1.70251 - 0.643964i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T \)
13 \( 1 + (3 - 2i)T \)
good3 \( 1 + (-1.61 + 1.61i)T - 3iT^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + (0.381 - 0.381i)T - 11iT^{2} \)
17 \( 1 + (2.23 - 2.23i)T - 17iT^{2} \)
19 \( 1 + (-0.854 + 0.854i)T - 19iT^{2} \)
23 \( 1 + (5.61 + 5.61i)T + 23iT^{2} \)
29 \( 1 + 0.763iT - 29T^{2} \)
31 \( 1 + (0.854 + 0.854i)T + 31iT^{2} \)
37 \( 1 + 3.70T + 37T^{2} \)
41 \( 1 + (-8.23 - 8.23i)T + 41iT^{2} \)
43 \( 1 + (2.85 + 2.85i)T + 43iT^{2} \)
47 \( 1 - 8.94T + 47T^{2} \)
53 \( 1 + (7.47 - 7.47i)T - 53iT^{2} \)
59 \( 1 + (-5.61 - 5.61i)T + 59iT^{2} \)
61 \( 1 + 7.70T + 61T^{2} \)
67 \( 1 + 13.7iT - 67T^{2} \)
71 \( 1 + (-0.381 - 0.381i)T + 71iT^{2} \)
73 \( 1 - 1.70iT - 73T^{2} \)
79 \( 1 - 9.70iT - 79T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 + (2.23 + 2.23i)T + 89iT^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−12.27868177196254686720474039336, −10.83308566652216353749128493584, −9.785939016390925166215852452276, −8.933379085424862841408939764380, −7.989356311657709949957245955978, −6.96986853803445877118360750435, −6.06633097091441290158688057823, −4.55758650333833254255346741928, −2.70127400932598440232406882825, −1.83097393338642605747821233270, 2.26583121904690600935967121760, 3.43786992834835518297032449083, 4.80405892166630396940819857740, 5.84768191231719152749060155318, 7.30789171909751043187156061535, 8.476605500887272880366826511410, 9.462093515704527581312392269241, 9.884836389483670839922463680163, 10.82622400550566886441045019075, 12.16355770520400609979029245092

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