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Properties

Label 2-735-21.5-c1-0-13
Degree $2$
Conductor $735$
Sign $0.997 - 0.0633i$
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  + (−1.5 − 0.866i)2-s + (1.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−1.5 − 2.59i)6-s + 1.73i·8-s + (1.5 + 2.59i)9-s + (−1.5 + 0.866i)10-s + (−3 + 1.73i)11-s + 1.73i·12-s − 3.46i·13-s + (1.5 − 0.866i)15-s + (2.49 − 4.33i)16-s + (3 + 5.19i)17-s − 5.19i·18-s + (6 + 3.46i)19-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (0.866 + 0.499i)3-s + (0.250 + 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.612 − 1.06i)6-s + 0.612i·8-s + (0.5 + 0.866i)9-s + (−0.474 + 0.273i)10-s + (−0.904 + 0.522i)11-s + 0.500i·12-s − 0.960i·13-s + (0.387 − 0.223i)15-s + (0.624 − 1.08i)16-s + (0.727 + 1.26i)17-s − 1.22i·18-s + (1.37 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.997 - 0.0633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17094 + 0.0371169i\)
\(L(\frac12)\) \(\approx\) \(1.17094 + 0.0371169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.73iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (3 - 1.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.12081795069356890849177740334, −9.762461707724518817279864473853, −8.791420692307502560887036280968, −7.958168204422630260044856493039, −7.66754596454278626410245235211, −5.70352292515603276330337282614, −4.96698557208501029033237445609, −3.52029523256780488347456354629, −2.49639712592843894223475676327, −1.30571291054086267471221576766, 0.915060577942144369755245303036, 2.58483747134084359262287939473, 3.52943232193821911354473291061, 5.13490071347792168220424589409, 6.47324495057860136401268846122, 7.22406830593214149349290596604, 7.70827584720661342318085965207, 8.652754806225204056131835372910, 9.380639126804241796461073885061, 9.861560244749447161958471116739

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