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Properties

Label 2-2535-195.134-c0-0-0
Degree $2$
Conductor $2535$
Sign $0.327 - 0.944i$
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  + (−1.56 − 0.900i)2-s + (−0.5 + 0.866i)3-s + (1.12 + 1.94i)4-s − i·5-s + (1.56 − 0.900i)6-s − 2.24i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.56i)10-s − 2.24·12-s + (0.866 + 0.5i)15-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s + 1.80i·18-s + (−1.07 + 0.623i)19-s + (1.94 − 1.12i)20-s + ⋯
L(s)  = 1  + (−1.56 − 0.900i)2-s + (−0.5 + 0.866i)3-s + (1.12 + 1.94i)4-s − i·5-s + (1.56 − 0.900i)6-s − 2.24i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.56i)10-s − 2.24·12-s + (0.866 + 0.5i)15-s + (−0.900 + 1.56i)16-s + (−0.222 − 0.385i)17-s + 1.80i·18-s + (−1.07 + 0.623i)19-s + (1.94 − 1.12i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ 0.327 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2316645297\)
\(L(\frac12)\) \(\approx\) \(0.2316645297\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 0.445iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.24iT - T^{2} \)
53 \( 1 - 1.24T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 - 0.445iT - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.256707956051617908265426900297, −8.802990952833049413549307935955, −8.132395532196258842341869362674, −7.31175232984037343051053620514, −6.15626730258219339663874547601, −5.29383424381103972504743730111, −4.21009741523429462823419402856, −3.53507987218481470530039046038, −2.25148252360124419258890154797, −1.15074310186457287185454314134, 0.29979611905697739328537391745, 1.87028287590656196407807934172, 2.57142395509000776120445572713, 4.33032700178151966780656633441, 5.73010022158014061801590258591, 6.18536500442901173258022221714, 6.92190568017350824896031871731, 7.22562029286785764262713256499, 8.339976336659427325356708416015, 8.452068848506020569656716922729

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