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Properties

Label 2-735-35.4-c1-0-28
Degree $2$
Conductor $735$
Sign $-0.796 + 0.604i$
Analytic cond. $5.86900$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.17 + 1.25i)2-s + (−0.866 − 0.5i)3-s + (2.16 − 3.75i)4-s + (−2.23 − 0.00136i)5-s + 2.51·6-s + 5.87i·8-s + (0.499 + 0.866i)9-s + (4.87 − 2.81i)10-s + (−0.489 + 0.847i)11-s + (−3.75 + 2.16i)12-s − 5.14i·13-s + (1.93 + 1.11i)15-s + (−3.05 − 5.29i)16-s + (3.59 + 2.07i)17-s + (−2.17 − 1.25i)18-s + (1.15 + 1.99i)19-s + ⋯
L(s)  = 1  + (−1.54 + 0.889i)2-s + (−0.499 − 0.288i)3-s + (1.08 − 1.87i)4-s + (−0.999 − 0.000610i)5-s + 1.02·6-s + 2.07i·8-s + (0.166 + 0.288i)9-s + (1.54 − 0.888i)10-s + (−0.147 + 0.255i)11-s + (−1.08 + 0.625i)12-s − 1.42i·13-s + (0.499 + 0.288i)15-s + (−0.763 − 1.32i)16-s + (0.871 + 0.503i)17-s + (−0.513 − 0.296i)18-s + (0.263 + 0.456i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.796 + 0.604i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.796 + 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0128814 - 0.0382462i\)
\(L(\frac12)\) \(\approx\) \(0.0128814 - 0.0382462i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (2.23 + 0.00136i)T \)
7 \( 1 \)
good2 \( 1 + (2.17 - 1.25i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (0.489 - 0.847i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.14iT - 13T^{2} \)
17 \( 1 + (-3.59 - 2.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.15 - 1.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.39 + 2.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 + (-0.316 + 0.548i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.84 - 4.52i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 0.344iT - 43T^{2} \)
47 \( 1 + (3.67 - 2.11i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.61 + 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.908 - 1.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.328 - 0.568i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.01 + 4.62i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.49T + 71T^{2} \)
73 \( 1 + (-4.65 - 2.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.44 + 9.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.62iT - 83T^{2} \)
89 \( 1 + (8.15 + 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.53iT - 97T^{2} \)
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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

−10.11580133612598303378139679401, −8.943530331191353602246069624434, −8.067854031315736259403821587889, −7.66568865776396280463505493289, −6.87785411699405941267562330810, −5.87351943648741293627026912148, −5.01632781556384013510551797553, −3.29358616837734707861359378556, −1.33552974800446800373202008377, −0.04276703862548397892690174578, 1.41731638685816548843917165969, 3.01026455036153057076930518015, 3.93695203102361649754448510667, 5.22876831208198344597885261759, 6.91096949808231588412225956791, 7.38616865407055393808629405323, 8.423451401153168102711195086669, 9.192961957108340422077839520493, 9.769118784062841425370323655186, 10.94674554228441162077037122940

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