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Properties

Label 2-735-15.8-c1-0-70
Degree $2$
Conductor $735$
Sign $-0.729 - 0.683i$
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  + (−0.929 − 0.929i)2-s + (1.05 − 1.37i)3-s − 0.272i·4-s + (−0.980 − 2.00i)5-s + (−2.25 + 0.299i)6-s + (−2.11 + 2.11i)8-s + (−0.782 − 2.89i)9-s + (−0.956 + 2.77i)10-s − 3.90i·11-s + (−0.374 − 0.287i)12-s + (1.56 + 1.56i)13-s + (−3.79 − 0.767i)15-s + 3.38·16-s + (−1.89 − 1.89i)17-s + (−1.96 + 3.41i)18-s + 1.86i·19-s + ⋯
L(s)  = 1  + (−0.657 − 0.657i)2-s + (0.607 − 0.794i)3-s − 0.136i·4-s + (−0.438 − 0.898i)5-s + (−0.921 + 0.122i)6-s + (−0.746 + 0.746i)8-s + (−0.260 − 0.965i)9-s + (−0.302 + 0.878i)10-s − 1.17i·11-s + (−0.108 − 0.0828i)12-s + (0.434 + 0.434i)13-s + (−0.980 − 0.198i)15-s + 0.845·16-s + (−0.459 − 0.459i)17-s + (−0.462 + 0.805i)18-s + 0.426i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.729 - 0.683i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324297 + 0.820111i\)
\(L(\frac12)\) \(\approx\) \(0.324297 + 0.820111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.05 + 1.37i)T \)
5 \( 1 + (0.980 + 2.00i)T \)
7 \( 1 \)
good2 \( 1 + (0.929 + 0.929i)T + 2iT^{2} \)
11 \( 1 + 3.90iT - 11T^{2} \)
13 \( 1 + (-1.56 - 1.56i)T + 13iT^{2} \)
17 \( 1 + (1.89 + 1.89i)T + 17iT^{2} \)
19 \( 1 - 1.86iT - 19T^{2} \)
23 \( 1 + (-1.74 + 1.74i)T - 23iT^{2} \)
29 \( 1 - 0.513T + 29T^{2} \)
31 \( 1 + 8.58T + 31T^{2} \)
37 \( 1 + (-4.83 + 4.83i)T - 37iT^{2} \)
41 \( 1 - 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (-3.74 - 3.74i)T + 47iT^{2} \)
53 \( 1 + (-1.36 + 1.36i)T - 53iT^{2} \)
59 \( 1 - 0.518T + 59T^{2} \)
61 \( 1 + 5.10T + 61T^{2} \)
67 \( 1 + (-6.40 + 6.40i)T - 67iT^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 + (-2.04 - 2.04i)T + 73iT^{2} \)
79 \( 1 - 5.05iT - 79T^{2} \)
83 \( 1 + (9.16 - 9.16i)T - 83iT^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (-6.81 + 6.81i)T - 97iT^{2} \)
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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

−9.448179364208245437309916278483, −9.065587640996263534976905674124, −8.378420030642986507558682417572, −7.60695744173552710471787052904, −6.31403301485855401553143205259, −5.48854842800096303021643655626, −4.00724566566620892995127067785, −2.83262187248806798883388874032, −1.57213345836091364154866117568, −0.53125673312800934254543366366, 2.43967296241279707026365865275, 3.52486693249549663068358751249, 4.30262714830814059760797417006, 5.73129319333986464519617174034, 6.99306932196703292321027997668, 7.45536477719979070178352430237, 8.351288311557237670348186472269, 9.094042399894873779522458430418, 9.899176145815512492336632653130, 10.64082533425082770469243402312

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