Properties

Label 2-735-105.59-c1-0-38
Degree $2$
Conductor $735$
Sign $-0.991 - 0.132i$
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  + (−0.866 − 1.5i)2-s + (−1.72 + 0.158i)3-s + (−0.5 + 0.866i)4-s + (−0.358 + 2.20i)5-s + (1.73 + 2.44i)6-s − 1.73·8-s + (2.94 − 0.548i)9-s + (3.62 − 1.37i)10-s + (−2.44 − 1.41i)11-s + (0.724 − 1.57i)12-s + 4·13-s + (0.267 − 3.86i)15-s + (2.49 + 4.33i)16-s + (2.44 + 1.41i)17-s + (−3.37 − 3.94i)18-s + ⋯
L(s)  = 1  + (−0.612 − 1.06i)2-s + (−0.995 + 0.0917i)3-s + (−0.250 + 0.433i)4-s + (−0.160 + 0.987i)5-s + (0.707 + 0.999i)6-s − 0.612·8-s + (0.983 − 0.182i)9-s + (1.14 − 0.434i)10-s + (−0.738 − 0.426i)11-s + (0.209 − 0.454i)12-s + 1.10·13-s + (0.0691 − 0.997i)15-s + (0.624 + 1.08i)16-s + (0.594 + 0.342i)17-s + (−0.795 − 0.930i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 - 0.132i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0190818 + 0.287356i\)
\(L(\frac12)\) \(\approx\) \(0.0190818 + 0.287356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.158i)T \)
5 \( 1 + (0.358 - 2.20i)T \)
7 \( 1 \)
good2 \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-2.44 - 1.41i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + (8.48 + 4.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 + (2.44 - 1.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.46 - 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.48 - 4.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.24 + 2.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.37144916664067983096958588232, −9.490964096286612736812285623309, −8.346841108646959613776324622290, −7.36709744444686206666642284800, −6.13190192212301205667842665594, −5.77366601816153655355051596720, −4.07180543743416624306869501127, −3.13166526079160935753840658232, −1.80217080287764123918181767626, −0.21897463438967241997858508972, 1.37401140284573142477393359906, 3.59499619455327732309988971279, 5.04259632714305600353730510794, 5.52981807019786811092677509006, 6.49767341169123190846437728276, 7.42440547561918341267882160203, 8.057049398694372611716665276317, 8.988781246750450093113768527151, 9.746160578300436942312949799762, 10.77324526749531586876513428426

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