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Properties

Label 2-260-20.7-c1-0-0
Degree $2$
Conductor $260$
Sign $0.371 - 0.928i$
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  + (−1.34 − 0.421i)2-s + (−1.90 − 1.90i)3-s + (1.64 + 1.13i)4-s + (−0.987 − 2.00i)5-s + (1.77 + 3.38i)6-s + (−3.15 + 3.15i)7-s + (−1.73 − 2.23i)8-s + 4.28i·9-s + (0.485 + 3.12i)10-s − 1.76i·11-s + (−0.964 − 5.31i)12-s + (0.707 − 0.707i)13-s + (5.58 − 2.92i)14-s + (−1.94 + 5.71i)15-s + (1.40 + 3.74i)16-s + (2.56 + 2.56i)17-s + ⋯
L(s)  = 1  + (−0.954 − 0.298i)2-s + (−1.10 − 1.10i)3-s + (0.822 + 0.569i)4-s + (−0.441 − 0.897i)5-s + (0.723 + 1.38i)6-s + (−1.19 + 1.19i)7-s + (−0.614 − 0.788i)8-s + 1.42i·9-s + (0.153 + 0.988i)10-s − 0.533i·11-s + (−0.278 − 1.53i)12-s + (0.196 − 0.196i)13-s + (1.49 − 0.781i)14-s + (−0.502 + 1.47i)15-s + (0.351 + 0.936i)16-s + (0.622 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.371 - 0.928i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.371 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111067 + 0.0752015i\)
\(L(\frac12)\) \(\approx\) \(0.111067 + 0.0752015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.421i)T \)
5 \( 1 + (0.987 + 2.00i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (1.90 + 1.90i)T + 3iT^{2} \)
7 \( 1 + (3.15 - 3.15i)T - 7iT^{2} \)
11 \( 1 + 1.76iT - 11T^{2} \)
17 \( 1 + (-2.56 - 2.56i)T + 17iT^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 + (-2.73 - 2.73i)T + 23iT^{2} \)
29 \( 1 - 2.75iT - 29T^{2} \)
31 \( 1 - 4.10iT - 31T^{2} \)
37 \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + (0.431 + 0.431i)T + 43iT^{2} \)
47 \( 1 + (-8.54 + 8.54i)T - 47iT^{2} \)
53 \( 1 + (10.2 - 10.2i)T - 53iT^{2} \)
59 \( 1 + 7.03T + 59T^{2} \)
61 \( 1 + 8.24T + 61T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (5.51 - 5.51i)T - 73iT^{2} \)
79 \( 1 + 6.65T + 79T^{2} \)
83 \( 1 + (3.92 + 3.92i)T + 83iT^{2} \)
89 \( 1 + 8.02iT - 89T^{2} \)
97 \( 1 + (-1.20 - 1.20i)T + 97iT^{2} \)
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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

−12.16084522033474370750034277055, −11.47541608600217258429876347038, −10.35663582368460174837555370081, −9.106233022764092715205310178094, −8.417736418805090909862156197642, −7.26625618839722920506840874016, −6.23568891935358668672722984334, −5.50904723096633959441106975004, −3.22653165728225866435590829352, −1.40048552394349729366520418402, 0.16929291466187613798202231580, 3.24391499845454324080243919342, 4.52034592691101558463536256638, 6.08217165606658549785169048517, 6.78158033577888641802506466673, 7.67468313336315939292273037376, 9.421311279025698287862977564580, 10.03038577062575787104740076756, 10.64114331462891779402868679266, 11.30176460671509131969074677631

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