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Properties

Label 2-260-20.7-c1-0-23
Degree $2$
Conductor $260$
Sign $0.973 - 0.229i$
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  + (−0.505 + 1.32i)2-s + (0.658 + 0.658i)3-s + (−1.48 − 1.33i)4-s + (0.820 − 2.07i)5-s + (−1.20 + 0.536i)6-s + (1.89 − 1.89i)7-s + (2.51 − 1.28i)8-s − 2.13i·9-s + (2.33 + 2.13i)10-s − 1.63i·11-s + (−0.0997 − 1.85i)12-s + (−0.707 + 0.707i)13-s + (1.54 + 3.45i)14-s + (1.90 − 0.828i)15-s + (0.428 + 3.97i)16-s + (2.78 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.357 + 0.933i)2-s + (0.379 + 0.379i)3-s + (−0.743 − 0.668i)4-s + (0.367 − 0.930i)5-s + (−0.490 + 0.218i)6-s + (0.714 − 0.714i)7-s + (0.890 − 0.455i)8-s − 0.711i·9-s + (0.737 + 0.675i)10-s − 0.493i·11-s + (−0.0287 − 0.536i)12-s + (−0.196 + 0.196i)13-s + (0.411 + 0.923i)14-s + (0.492 − 0.213i)15-s + (0.107 + 0.994i)16-s + (0.674 + 0.674i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.973 - 0.229i$
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: \((2,\ 260,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21927 + 0.141640i\)
\(L(\frac12)\) \(\approx\) \(1.21927 + 0.141640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.505 - 1.32i)T \)
5 \( 1 + (-0.820 + 2.07i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.658 - 0.658i)T + 3iT^{2} \)
7 \( 1 + (-1.89 + 1.89i)T - 7iT^{2} \)
11 \( 1 + 1.63iT - 11T^{2} \)
17 \( 1 + (-2.78 - 2.78i)T + 17iT^{2} \)
19 \( 1 + 3.52T + 19T^{2} \)
23 \( 1 + (0.868 + 0.868i)T + 23iT^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 - 7.80iT - 31T^{2} \)
37 \( 1 + (-4.02 - 4.02i)T + 37iT^{2} \)
41 \( 1 - 5.49T + 41T^{2} \)
43 \( 1 + (5.16 + 5.16i)T + 43iT^{2} \)
47 \( 1 + (-8.12 + 8.12i)T - 47iT^{2} \)
53 \( 1 + (0.551 - 0.551i)T - 53iT^{2} \)
59 \( 1 - 8.09T + 59T^{2} \)
61 \( 1 + 8.43T + 61T^{2} \)
67 \( 1 + (1.32 - 1.32i)T - 67iT^{2} \)
71 \( 1 - 6.50iT - 71T^{2} \)
73 \( 1 + (4.95 - 4.95i)T - 73iT^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + (-6.57 - 6.57i)T + 83iT^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + (0.0106 + 0.0106i)T + 97iT^{2} \)
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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

−12.19685514223921232065092234326, −10.66307050526341765223948091797, −9.939810812735016060392941842701, −8.761139830246828263259741760892, −8.463425279023605228824900773815, −7.16415975633914989555714308639, −5.98089396343279907700084979677, −4.86836574933559323149736676361, −3.89488621498352667283099230041, −1.22168165305285123120545359242, 2.03270922744866953076103109400, 2.71307087072773745901559556812, 4.43329234980130386084007371012, 5.79009638202530968830504154878, 7.50588955675265871448442485524, 7.992491604562101840424616824964, 9.272006311432950000087331772245, 10.09979767057635329008829319587, 11.06835411303910594916636816684, 11.75214113872075906027130683972

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