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Properties

Label 2-260-260.103-c1-0-3
Degree $2$
Conductor $260$
Sign $-0.649 - 0.760i$
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  + (−1.31 + 0.528i)2-s + (−0.834 + 0.834i)3-s + (1.44 − 1.38i)4-s + (−0.462 − 2.18i)5-s + (0.653 − 1.53i)6-s + (−1.21 + 1.21i)7-s + (−1.15 + 2.58i)8-s + 1.60i·9-s + (1.76 + 2.62i)10-s + 0.135·11-s + (−0.0453 + 2.35i)12-s + (1.01 + 3.45i)13-s + (0.948 − 2.22i)14-s + (2.21 + 1.43i)15-s + (0.153 − 3.99i)16-s + (−1.10 + 1.10i)17-s + ⋯
L(s)  = 1  + (−0.927 + 0.373i)2-s + (−0.481 + 0.481i)3-s + (0.720 − 0.693i)4-s + (−0.206 − 0.978i)5-s + (0.266 − 0.626i)6-s + (−0.457 + 0.457i)7-s + (−0.409 + 0.912i)8-s + 0.535i·9-s + (0.557 + 0.830i)10-s + 0.0409·11-s + (−0.0130 + 0.681i)12-s + (0.282 + 0.959i)13-s + (0.253 − 0.595i)14-s + (0.570 + 0.371i)15-s + (0.0384 − 0.999i)16-s + (−0.269 + 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.649 - 0.760i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188594 + 0.409163i\)
\(L(\frac12)\) \(\approx\) \(0.188594 + 0.409163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.528i)T \)
5 \( 1 + (0.462 + 2.18i)T \)
13 \( 1 + (-1.01 - 3.45i)T \)
good3 \( 1 + (0.834 - 0.834i)T - 3iT^{2} \)
7 \( 1 + (1.21 - 1.21i)T - 7iT^{2} \)
11 \( 1 - 0.135T + 11T^{2} \)
17 \( 1 + (1.10 - 1.10i)T - 17iT^{2} \)
19 \( 1 - 6.12iT - 19T^{2} \)
23 \( 1 + (4.72 - 4.72i)T - 23iT^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 + (-6.39 - 6.39i)T + 37iT^{2} \)
41 \( 1 - 2.92iT - 41T^{2} \)
43 \( 1 + (1.51 - 1.51i)T - 43iT^{2} \)
47 \( 1 + (-8.14 + 8.14i)T - 47iT^{2} \)
53 \( 1 + (5.39 + 5.39i)T + 53iT^{2} \)
59 \( 1 - 5.47iT - 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + (-7.10 + 7.10i)T - 67iT^{2} \)
71 \( 1 + 4.49T + 71T^{2} \)
73 \( 1 + (6.04 - 6.04i)T - 73iT^{2} \)
79 \( 1 + 0.944T + 79T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 + 3.22T + 89T^{2} \)
97 \( 1 + (2.31 + 2.31i)T + 97iT^{2} \)
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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

−11.90007050436092848570011406454, −11.42149890963162555883070280576, −10.11508491675629520414740331458, −9.534529865331525738259573178375, −8.495755277145215349980265373257, −7.72185439552596414261128085200, −6.20173166719648610520498262057, −5.47153938632543753051743262162, −4.13707011359158620472255008656, −1.83104445143174542992157907489, 0.49006064439949305844276878837, 2.63056534206117277717796181201, 3.80115929924293993421322801367, 6.04222641433097691454932829150, 6.88259756322980490856896691985, 7.53483935038081878662770111246, 8.835448095798690380484122257135, 9.894128423901336309923287506809, 10.79824949285578327762033914291, 11.34668326116819728612382017815

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