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Properties

Label 2-2535-195.128-c0-0-0
Degree $2$
Conductor $2535$
Sign $-0.563 - 0.826i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 + 0.923i)2-s + (−0.258 + 0.965i)3-s + (1.20 − 2.09i)4-s + (−0.923 + 0.382i)5-s + (−0.478 − 1.78i)6-s + 2.61i·8-s + (−0.866 − 0.499i)9-s + (1.12 − 1.46i)10-s + (0.198 − 0.739i)11-s + (1.70 + 1.70i)12-s + (−0.130 − 0.991i)15-s + (−1.20 − 2.09i)16-s + 1.84·18-s + (−0.315 + 2.39i)20-s + (0.366 + 1.36i)22-s + ⋯
L(s)  = 1  + (−1.60 + 0.923i)2-s + (−0.258 + 0.965i)3-s + (1.20 − 2.09i)4-s + (−0.923 + 0.382i)5-s + (−0.478 − 1.78i)6-s + 2.61i·8-s + (−0.866 − 0.499i)9-s + (1.12 − 1.46i)10-s + (0.198 − 0.739i)11-s + (1.70 + 1.70i)12-s + (−0.130 − 0.991i)15-s + (−1.20 − 2.09i)16-s + 1.84·18-s + (−0.315 + 2.39i)20-s + (0.366 + 1.36i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ -0.563 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3616509003\)
\(L(\frac12)\) \(\approx\) \(0.3616509003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 \)
good2 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.198 + 0.739i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 - 0.765T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.478 + 1.78i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.478 - 1.78i)T + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - 0.765T + T^{2} \)
89 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.282452029970322381465014339464, −8.534080404853350577618552704543, −8.034140084007169690673907205392, −7.24847700408866689987500759202, −6.37955259863795468186348598492, −5.83512237390253488958619859198, −4.79966703361030334390242376631, −3.77683663602920279406245847430, −2.67404330312261990075029842659, −0.836426180446719018377168378336, 0.63041861138573426409709580112, 1.68119409184863618068257485039, 2.58969440352461624058189791332, 3.61732653921991521365849238538, 4.69679000689910223011555794141, 6.05630461005371785278912345288, 7.09682944499629240489710087464, 7.56219975266670466729646666321, 8.027027165563165336147354047576, 9.007076528540877879277603803377

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