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Properties

Label 2-260-65.37-c1-0-1
Degree $2$
Conductor $260$
Sign $-0.172 - 0.985i$
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  + (−1.44 − 0.387i)3-s + (1.07 + 1.95i)5-s + (−1.21 + 2.10i)7-s + (−0.655 − 0.378i)9-s + (−2.60 − 0.697i)11-s + (0.447 + 3.57i)13-s + (−0.803 − 3.25i)15-s + (1.62 + 6.05i)17-s + (1.00 + 3.73i)19-s + (2.57 − 2.57i)21-s + (−0.307 + 1.14i)23-s + (−2.66 + 4.22i)25-s + (3.97 + 3.97i)27-s + (5.29 − 3.05i)29-s + (−6.22 − 6.22i)31-s + ⋯
L(s)  = 1  + (−0.835 − 0.223i)3-s + (0.482 + 0.875i)5-s + (−0.459 + 0.795i)7-s + (−0.218 − 0.126i)9-s + (−0.784 − 0.210i)11-s + (0.123 + 0.992i)13-s + (−0.207 − 0.839i)15-s + (0.393 + 1.46i)17-s + (0.229 + 0.856i)19-s + (0.561 − 0.561i)21-s + (−0.0641 + 0.239i)23-s + (−0.533 + 0.845i)25-s + (0.765 + 0.765i)27-s + (0.982 − 0.567i)29-s + (−1.11 − 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.172 - 0.985i$
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: \((2,\ 260,\ (\ :1/2),\ -0.172 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482648 + 0.574500i\)
\(L(\frac12)\) \(\approx\) \(0.482648 + 0.574500i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 - 1.95i)T \)
13 \( 1 + (-0.447 - 3.57i)T \)
good3 \( 1 + (1.44 + 0.387i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.21 - 2.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.60 + 0.697i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.62 - 6.05i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.00 - 3.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.307 - 1.14i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.29 + 3.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.22 + 6.22i)T + 31iT^{2} \)
37 \( 1 + (3.86 + 6.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.913 + 3.41i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.37 + 1.17i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 6.81T + 47T^{2} \)
53 \( 1 + (-4.88 + 4.88i)T - 53iT^{2} \)
59 \( 1 + (-9.04 + 2.42i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.87 - 8.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.878 - 0.507i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.5 + 2.83i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 9.06iT - 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + 1.09T + 83T^{2} \)
89 \( 1 + (-1.29 + 4.83i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.39 - 1.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.18520689325108827239822479779, −11.29573133213691836463224594231, −10.49989199125195818398724549733, −9.552307797125758806726475858488, −8.388255848246524997338305562516, −7.05643824591278256382613307929, −5.95596831023596188405256830931, −5.68059490321960928588166724036, −3.65780928843996221859071601354, −2.18526027034352268341573186961, 0.63524675331788176180096550376, 2.98000900215868198914420920943, 4.89246021455890110004296121219, 5.27407292005851734331218280822, 6.60345492718762715745061452442, 7.77251816063827583107525300988, 8.939781852309358553236184993797, 10.07379389380713654485165104257, 10.61477073096730875816088319764, 11.73931676148869753587243534529

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