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Properties

Label 2-735-105.104-c1-0-22
Degree $2$
Conductor $735$
Sign $-0.0804 - 0.996i$
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  − 1.51·2-s + (1.66 + 0.476i)3-s + 0.294·4-s + (−0.775 + 2.09i)5-s + (−2.52 − 0.722i)6-s + 2.58·8-s + (2.54 + 1.58i)9-s + (1.17 − 3.17i)10-s + 2.14i·11-s + (0.490 + 0.140i)12-s + 3.48·13-s + (−2.29 + 3.12i)15-s − 4.50·16-s − 3.57i·17-s + (−3.85 − 2.40i)18-s + 1.22i·19-s + ⋯
L(s)  = 1  − 1.07·2-s + (0.961 + 0.275i)3-s + 0.147·4-s + (−0.346 + 0.937i)5-s + (−1.02 − 0.294i)6-s + 0.913·8-s + (0.848 + 0.529i)9-s + (0.371 − 1.00i)10-s + 0.647i·11-s + (0.141 + 0.0405i)12-s + 0.965·13-s + (−0.591 + 0.806i)15-s − 1.12·16-s − 0.867i·17-s + (−0.908 − 0.566i)18-s + 0.280i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0804 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0804 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0804 - 0.996i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.0804 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709969 + 0.769554i\)
\(L(\frac12)\) \(\approx\) \(0.709969 + 0.769554i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 - 0.476i)T \)
5 \( 1 + (0.775 - 2.09i)T \)
7 \( 1 \)
good2 \( 1 + 1.51T + 2T^{2} \)
11 \( 1 - 2.14iT - 11T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + 3.57iT - 17T^{2} \)
19 \( 1 - 1.22iT - 19T^{2} \)
23 \( 1 - 1.51T + 23T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 - 3.17iT - 31T^{2} \)
37 \( 1 - 7.80iT - 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 2.99iT - 43T^{2} \)
47 \( 1 + 6.10iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 - 3.39iT - 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 6.85T + 73T^{2} \)
79 \( 1 + 1.88T + 79T^{2} \)
83 \( 1 - 9.10iT - 83T^{2} \)
89 \( 1 + 1.77T + 89T^{2} \)
97 \( 1 + 1.32T + 97T^{2} \)
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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

−10.26529595286694030537587994499, −9.811179403788908526889733250931, −8.760025235587054494370638965000, −8.312423964422636154732159163061, −7.26622265293656629983548242752, −6.83643291975848273394962596910, −5.02367890864129727487730476497, −3.92809642289527847744032102588, −2.93616149110330845871843293530, −1.57470053968519474374423006858, 0.76182169767505530256928210376, 1.89178170974177273436247092421, 3.60676655505825585210993113989, 4.39991379587586441996695389195, 5.81755564422977157199527055167, 7.09555304593212965549151040396, 8.045181883223702878197008746175, 8.493676063608862729144248833530, 9.023450887261831233230651152374, 9.830835627837044547166691924763

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