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Properties

Label 2-2535-195.128-c0-0-3
Degree $2$
Conductor $2535$
Sign $-0.826 + 0.563i$
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  + (−0.662 + 0.382i)2-s + (0.258 − 0.965i)3-s + (−0.207 + 0.358i)4-s + (−0.382 − 0.923i)5-s + (0.198 + 0.739i)6-s − 1.08i·8-s + (−0.866 − 0.499i)9-s + (0.607 + 0.465i)10-s + (0.478 − 1.78i)11-s + (0.292 + 0.292i)12-s + (−0.991 + 0.130i)15-s + (0.207 + 0.358i)16-s + 0.765·18-s + (0.410 + 0.0540i)20-s + (0.366 + 1.36i)22-s + ⋯
L(s)  = 1  + (−0.662 + 0.382i)2-s + (0.258 − 0.965i)3-s + (−0.207 + 0.358i)4-s + (−0.382 − 0.923i)5-s + (0.198 + 0.739i)6-s − 1.08i·8-s + (−0.866 − 0.499i)9-s + (0.607 + 0.465i)10-s + (0.478 − 1.78i)11-s + (0.292 + 0.292i)12-s + (−0.991 + 0.130i)15-s + (0.207 + 0.358i)16-s + 0.765·18-s + (0.410 + 0.0540i)20-s + (0.366 + 1.36i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.826 + 0.563i$
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: \((2,\ 2535,\ (\ :0),\ -0.826 + 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5514422580\)
\(L(\frac12)\) \(\approx\) \(0.5514422580\)
\(L(1)\) 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 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + 1.84T + T^{2} \)
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 \( 1 + 1.84T + T^{2} \)
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

−8.587185903667229971464090503395, −8.168666390319439806652976712659, −7.54534764639762995096494303809, −6.58845787937855503914176670198, −5.99439500280915654540178483756, −4.90969111255992269768695756968, −3.70857285395053698960921529834, −3.14018053792819557825033080831, −1.47632722132144123842882845547, −0.45799584917068910326926614036, 1.82869545229452972235002609080, 2.67766743602028354300916374192, 3.79506678359866602034136450926, 4.56131817349510093983941657558, 5.30511012634980823969481082444, 6.40057706384836181439997378609, 7.26198286508961606662101455831, 8.073743481006201523678986119600, 8.852631572420697496909999683564, 9.628294181841669790443455516193

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