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Properties

Label 2-2535-195.167-c0-0-0
Degree $2$
Conductor $2535$
Sign $0.605 - 0.795i$
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  + (0.382 + 0.662i)2-s + (−0.965 − 0.258i)3-s + (0.207 − 0.358i)4-s + (−0.923 − 0.382i)5-s + (−0.198 − 0.739i)6-s + 1.08·8-s + (0.866 + 0.499i)9-s + (−0.0999 − 0.758i)10-s + (−0.478 + 1.78i)11-s + (−0.292 + 0.292i)12-s + (0.793 + 0.608i)15-s + (0.207 + 0.358i)16-s + 0.765i·18-s + (−0.328 + 0.252i)20-s + (−1.36 + 0.366i)22-s + ⋯
L(s)  = 1  + (0.382 + 0.662i)2-s + (−0.965 − 0.258i)3-s + (0.207 − 0.358i)4-s + (−0.923 − 0.382i)5-s + (−0.198 − 0.739i)6-s + 1.08·8-s + (0.866 + 0.499i)9-s + (−0.0999 − 0.758i)10-s + (−0.478 + 1.78i)11-s + (−0.292 + 0.292i)12-s + (0.793 + 0.608i)15-s + (0.207 + 0.358i)16-s + 0.765i·18-s + (−0.328 + 0.252i)20-s + (−1.36 + 0.366i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (2117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9896815337\)
\(L(\frac12)\) \(\approx\) \(0.9896815337\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 \)
good2 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 - 1.84iT - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + 1.84iT - T^{2} \)
89 \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \)
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

−9.288145565708815655009116604678, −7.922907415686705691112773976825, −7.47666585932682192611196343001, −6.96829654498124097664521248594, −6.09500220424600979230683875228, −5.28728128840990796186119353392, −4.60097380649052907905101795318, −4.16137753603146557133977277697, −2.33727679145776785121565548687, −1.17443276445380987543862022229, 0.77815443932410728356722242794, 2.47928507093160454714166301421, 3.49477785018378501788926624223, 3.91658734409528131586535630220, 4.92755972035424530392242711630, 5.76992212317912776217938743107, 6.64794832298857442627765304431, 7.42418020476355371680037868668, 8.125274127320525854356806039272, 8.917762886850713369821558719040

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