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Properties

Label 2-260-20.3-c1-0-19
Degree $2$
Conductor $260$
Sign $0.738 - 0.673i$
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  + (−0.421 + 1.34i)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 + (2.23 − 1.73i)8-s − 4.28i·9-s + (−2.29 − 2.17i)10-s − 1.76i·11-s + (−5.31 + 0.964i)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  + (−0.298 + 0.954i)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 + (0.788 − 0.614i)8-s − 1.42i·9-s + (−0.724 − 0.689i)10-s − 0.533i·11-s + (−1.53 + 0.278i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.738 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37244 + 0.531993i\)
\(L(\frac12)\) \(\approx\) \(1.37244 + 0.531993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.421 - 1.34i)T \)
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

−12.11233069962182770196775843575, −11.33574201413479424997696144040, −9.825979734095312981341206502468, −8.616760692432569086267291489897, −8.167664768497557218213881760228, −7.40606583807430180266604158188, −6.42793310381778700295628777320, −5.21549234746010210038384158722, −3.33716351902619654907321475775, −1.84026423655313947734310356150, 1.55016955213805251086938340694, 3.41993995366238740625201755326, 4.32556552315419593551865206882, 4.87455758614626488825450475909, 7.81924167276493466215860618245, 8.096978126077074616165203544941, 9.118999574433749660302985399474, 10.02157198246318726414785577175, 10.69152010830608949451383272350, 11.68614061413785850421830875466

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