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Properties

Label 2-260-65.37-c1-0-2
Degree $2$
Conductor $260$
Sign $0.624 - 0.781i$
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  + (2.10 + 0.563i)3-s + (−1.38 + 1.75i)5-s + (−0.724 + 1.25i)7-s + (1.50 + 0.867i)9-s + (4.39 + 1.17i)11-s + (1.55 + 3.25i)13-s + (−3.89 + 2.91i)15-s + (−1.39 − 5.20i)17-s + (−0.379 − 1.41i)19-s + (−2.23 + 2.23i)21-s + (0.968 − 3.61i)23-s + (−1.18 − 4.85i)25-s + (−1.94 − 1.94i)27-s + (−0.251 + 0.145i)29-s + (−1.02 − 1.02i)31-s + ⋯
L(s)  = 1  + (1.21 + 0.325i)3-s + (−0.617 + 0.786i)5-s + (−0.273 + 0.474i)7-s + (0.500 + 0.289i)9-s + (1.32 + 0.354i)11-s + (0.431 + 0.902i)13-s + (−1.00 + 0.753i)15-s + (−0.338 − 1.26i)17-s + (−0.0870 − 0.324i)19-s + (−0.486 + 0.486i)21-s + (0.202 − 0.753i)23-s + (−0.237 − 0.971i)25-s + (−0.374 − 0.374i)27-s + (−0.0467 + 0.0269i)29-s + (−0.184 − 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.624 - 0.781i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.624 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49425 + 0.718430i\)
\(L(\frac12)\) \(\approx\) \(1.49425 + 0.718430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 1.75i)T \)
13 \( 1 + (-1.55 - 3.25i)T \)
good3 \( 1 + (-2.10 - 0.563i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.724 - 1.25i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.39 - 1.17i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.39 + 5.20i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.379 + 1.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.968 + 3.61i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.251 - 0.145i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.02 + 1.02i)T + 31iT^{2} \)
37 \( 1 + (-3.59 - 6.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.53 + 9.46i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.82 - 2.36i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 4.46T + 47T^{2} \)
53 \( 1 + (-3.60 + 3.60i)T - 53iT^{2} \)
59 \( 1 + (-5.41 + 1.45i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.00 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.28 - 0.739i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 - 2.96i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 + 5.42iT - 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 + (-3.62 + 13.5i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.33 - 5.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.84966305043767543173033469966, −11.39767689322087900622410980769, −9.946138229203666956190801104217, −9.134836172556384278200717408444, −8.498938041921451849608323679087, −7.17555703719251307301470075831, −6.43826432330814513338265712072, −4.44954948551591308137379012414, −3.48728216749049736346141735153, −2.39105981227935769694883326538, 1.41988581486983866232886608834, 3.37809875021786561510254077375, 4.08174014722812267245009460663, 5.83222329327981911843307261745, 7.17713167360049938933963040636, 8.208172609959096079211794732623, 8.691038063748626499202667270545, 9.663304376363075976018270474603, 10.96382619944364006000687328298, 11.98824459369256534548720485335

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