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Properties

Label 2-260-65.7-c1-0-6
Degree $2$
Conductor $260$
Sign $-0.999 + 0.0318i$
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  + (−0.668 − 2.49i)3-s + (−2.20 − 0.380i)5-s + (−0.432 − 0.749i)7-s + (−3.17 + 1.83i)9-s + (0.417 + 1.55i)11-s + (−3.35 − 1.32i)13-s + (0.522 + 5.74i)15-s + (−4.99 − 1.33i)17-s + (1.16 + 0.312i)19-s + (−1.57 + 1.57i)21-s + (2.62 − 0.704i)23-s + (4.70 + 1.67i)25-s + (1.20 + 1.20i)27-s + (−5.94 − 3.43i)29-s + (0.191 + 0.191i)31-s + ⋯
L(s)  = 1  + (−0.385 − 1.43i)3-s + (−0.985 − 0.170i)5-s + (−0.163 − 0.283i)7-s + (−1.05 + 0.610i)9-s + (0.125 + 0.469i)11-s + (−0.930 − 0.367i)13-s + (0.134 + 1.48i)15-s + (−1.21 − 0.324i)17-s + (0.267 + 0.0716i)19-s + (−0.344 + 0.344i)21-s + (0.548 − 0.146i)23-s + (0.941 + 0.335i)25-s + (0.232 + 0.232i)27-s + (−1.10 − 0.636i)29-s + (0.0344 + 0.0344i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.999 + 0.0318i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.999 + 0.0318i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00880072 - 0.551937i\)
\(L(\frac12)\) \(\approx\) \(0.00880072 - 0.551937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.20 + 0.380i)T \)
13 \( 1 + (3.35 + 1.32i)T \)
good3 \( 1 + (0.668 + 2.49i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.432 + 0.749i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.417 - 1.55i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (4.99 + 1.33i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.16 - 0.312i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.62 + 0.704i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (5.94 + 3.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.191 - 0.191i)T + 31iT^{2} \)
37 \( 1 + (-4.74 + 8.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.417 + 0.111i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.09 + 11.5i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 + (4.88 - 4.88i)T - 53iT^{2} \)
59 \( 1 + (-3.05 + 11.3i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.94 + 5.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.7 - 6.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.85 - 14.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 - 3.98iT - 79T^{2} \)
83 \( 1 + 6.25T + 83T^{2} \)
89 \( 1 + (-9.80 + 2.62i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.26 + 0.728i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.66388470411273601189372232372, −10.97513211134842152306288633506, −9.512451139813450020018237371885, −8.287099715429763742774499192018, −7.25024919263929027620938258915, −6.98939824604150750064523117997, −5.48017008415786783747568304319, −4.12210894008085543427403959310, −2.32201908870385757946918123067, −0.43833812443299429942655918249, 3.05206785330400233443794790989, 4.22414181672390861091653830628, 4.97465099922132350486736438017, 6.35500803858669621735875959393, 7.62525242959550687396674331361, 8.907889128774002227287505551014, 9.565296182704585638306805059639, 10.77495008135333232465378590126, 11.25724378464820057074787847521, 12.13027370351405368088157895235

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