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Properties

Label 2-260-260.207-c1-0-21
Degree $2$
Conductor $260$
Sign $-0.339 + 0.940i$
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  + (−1.23 + 0.693i)2-s + (−0.828 − 0.828i)3-s + (1.03 − 1.70i)4-s + (−1.31 − 1.80i)5-s + (1.59 + 0.446i)6-s + (2.94 + 2.94i)7-s + (−0.0923 + 2.82i)8-s − 1.62i·9-s + (2.87 + 1.30i)10-s − 3.40·11-s + (−2.27 + 0.556i)12-s + (−1.99 − 3.00i)13-s + (−5.66 − 1.58i)14-s + (−0.402 + 2.58i)15-s + (−1.84 − 3.54i)16-s + (−2.96 − 2.96i)17-s + ⋯
L(s)  = 1  + (−0.871 + 0.490i)2-s + (−0.478 − 0.478i)3-s + (0.518 − 0.854i)4-s + (−0.590 − 0.807i)5-s + (0.651 + 0.182i)6-s + (1.11 + 1.11i)7-s + (−0.0326 + 0.999i)8-s − 0.542i·9-s + (0.910 + 0.414i)10-s − 1.02·11-s + (−0.656 + 0.160i)12-s + (−0.552 − 0.833i)13-s + (−1.51 − 0.423i)14-s + (−0.103 + 0.668i)15-s + (−0.461 − 0.886i)16-s + (−0.719 − 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.339 + 0.940i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.339 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.268811 - 0.382812i\)
\(L(\frac12)\) \(\approx\) \(0.268811 - 0.382812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.693i)T \)
5 \( 1 + (1.31 + 1.80i)T \)
13 \( 1 + (1.99 + 3.00i)T \)
good3 \( 1 + (0.828 + 0.828i)T + 3iT^{2} \)
7 \( 1 + (-2.94 - 2.94i)T + 7iT^{2} \)
11 \( 1 + 3.40T + 11T^{2} \)
17 \( 1 + (2.96 + 2.96i)T + 17iT^{2} \)
19 \( 1 + 7.33iT - 19T^{2} \)
23 \( 1 + (-1.65 - 1.65i)T + 23iT^{2} \)
29 \( 1 + 1.98iT - 29T^{2} \)
31 \( 1 + 4.43T + 31T^{2} \)
37 \( 1 + (-1.88 + 1.88i)T - 37iT^{2} \)
41 \( 1 + 7.55iT - 41T^{2} \)
43 \( 1 + (-0.548 - 0.548i)T + 43iT^{2} \)
47 \( 1 + (5.58 + 5.58i)T + 47iT^{2} \)
53 \( 1 + (0.437 - 0.437i)T - 53iT^{2} \)
59 \( 1 - 6.14iT - 59T^{2} \)
61 \( 1 - 8.12T + 61T^{2} \)
67 \( 1 + (-3.78 - 3.78i)T + 67iT^{2} \)
71 \( 1 - 8.90T + 71T^{2} \)
73 \( 1 + (-4.20 - 4.20i)T + 73iT^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + (0.688 - 0.688i)T - 83iT^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (3.63 - 3.63i)T - 97iT^{2} \)
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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

−11.52872942286243473080746192762, −11.00258263925892552631035401969, −9.410523024085175566674575519534, −8.724858617474228105656879263381, −7.83409709081385328231737062722, −6.99898385496930966818841337965, −5.42128656443919921205684316237, −5.06627462909974047919701846296, −2.35695594652149723470277505641, −0.49506960848957732323250687056, 2.02725156100717459431274891057, 3.79273450411814582949833001450, 4.77377289733053947479211254920, 6.61965146935936234273168959976, 7.81404328952439901260640927964, 8.047898717687090722409867359986, 9.813062583147819881240355904656, 10.73487251178194166511584057613, 10.87153786188879683427269367056, 11.77338943215927932513634628223

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