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Properties

Label 2-2535-195.158-c0-0-0
Degree $2$
Conductor $2535$
Sign $0.374 - 0.927i$
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  + (−0.923 − 1.60i)2-s + (−0.258 + 0.965i)3-s + (−1.20 + 2.09i)4-s + (−0.382 − 0.923i)5-s + (1.78 − 0.478i)6-s + 2.61·8-s + (−0.866 − 0.499i)9-s + (−1.12 + 1.46i)10-s + (−0.739 − 0.198i)11-s + (−1.70 − 1.70i)12-s + (0.991 − 0.130i)15-s + (−1.20 − 2.09i)16-s + 1.84i·18-s + (2.39 + 0.315i)20-s + (0.366 + 1.36i)22-s + ⋯
L(s)  = 1  + (−0.923 − 1.60i)2-s + (−0.258 + 0.965i)3-s + (−1.20 + 2.09i)4-s + (−0.382 − 0.923i)5-s + (1.78 − 0.478i)6-s + 2.61·8-s + (−0.866 − 0.499i)9-s + (−1.12 + 1.46i)10-s + (−0.739 − 0.198i)11-s + (−1.70 − 1.70i)12-s + (0.991 − 0.130i)15-s + (−1.20 − 2.09i)16-s + 1.84i·18-s + (2.39 + 0.315i)20-s + (0.366 + 1.36i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (2108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ 0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1270158125\)
\(L(\frac12)\) \(\approx\) \(0.1270158125\)
\(L(1)\) 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 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 \)
good2 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.739 + 0.198i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + 0.765iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - 0.765iT - T^{2} \)
89 \( 1 + (-0.198 + 0.739i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.397966703744314101820452119331, −8.699492518571628919438280554047, −8.274353847652695584505773091683, −7.40804262577695285103342658788, −5.84484863066972867182580607987, −4.89078072755656177248453489579, −4.26155092808430573327861381070, −3.42816988279784162586590927492, −2.60974053157987317063201655699, −1.24658234444572706464014116872, 0.12912812201237893800344139940, 1.76851823168561062156641969468, 3.05373393588266713145944208532, 4.61208023709061112504309684310, 5.50686144402019738474745295544, 6.22504496263551496037788371674, 6.77093604923719169756006454734, 7.56201765418785624466504599897, 7.78115963790232343094538348141, 8.598282025316409300875512151175

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