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Properties

Label 2-260-65.7-c1-0-4
Degree $2$
Conductor $260$
Sign $0.439 + 0.898i$
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  + (−0.5 − 1.86i)3-s + (2 − i)5-s + (1.86 + 3.23i)7-s + (−0.633 + 0.366i)9-s + (−0.598 − 2.23i)11-s + (2 − 3i)13-s + (−2.86 − 3.23i)15-s + (−4.23 − 1.13i)17-s + (−0.866 − 0.232i)19-s + (5.09 − 5.09i)21-s + (−6.96 + 1.86i)23-s + (3 − 4i)25-s + (−3.09 − 3.09i)27-s + (7.96 + 4.59i)29-s + (5.73 + 5.73i)31-s + ⋯
L(s)  = 1  + (−0.288 − 1.07i)3-s + (0.894 − 0.447i)5-s + (0.705 + 1.22i)7-s + (−0.211 + 0.122i)9-s + (−0.180 − 0.672i)11-s + (0.554 − 0.832i)13-s + (−0.740 − 0.834i)15-s + (−1.02 − 0.275i)17-s + (−0.198 − 0.0532i)19-s + (1.11 − 1.11i)21-s + (−1.45 + 0.389i)23-s + (0.600 − 0.800i)25-s + (−0.596 − 0.596i)27-s + (1.47 + 0.853i)29-s + (1.02 + 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.439 + 0.898i$
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: \((2,\ 260,\ (\ :1/2),\ 0.439 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17905 - 0.735752i\)
\(L(\frac12)\) \(\approx\) \(1.17905 - 0.735752i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
13 \( 1 + (-2 + 3i)T \)
good3 \( 1 + (0.5 + 1.86i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.86 - 3.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.598 + 2.23i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (4.23 + 1.13i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.866 + 0.232i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (6.96 - 1.86i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-7.96 - 4.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.73 - 5.73i)T + 31iT^{2} \)
37 \( 1 + (0.133 - 0.232i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.133 - 0.0358i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.03 - 11.3i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 0.535T + 47T^{2} \)
53 \( 1 + (1.53 - 1.53i)T - 53iT^{2} \)
59 \( 1 + (-1.79 + 6.69i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.79 + 2.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.13 + 4.23i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 - 4.53iT - 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (14.7 - 3.96i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.86 + 2.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.03968739827010015843423989159, −11.07848395949404193133330715065, −9.888007141786970491052706853371, −8.591950980239454467586998966182, −8.183443818970159487408686624807, −6.54454765080026417625230205035, −5.90594786496542345577679402283, −4.89362049532440699378101056381, −2.62721635842540548966459577979, −1.38225971131092144365299955548, 2.01387486736479624708741480889, 4.10798740537627876937749113685, 4.60535029183819381395175427330, 6.09366342440689977196556217738, 7.07951996508441562862395869197, 8.394313249203549423575060590861, 9.690069161634447336180570822693, 10.30778833293727322772625821597, 10.85170994167245621321609462200, 11.87391051026925772277639525917

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