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Properties

Label 2-735-15.2-c1-0-14
Degree $2$
Conductor $735$
Sign $0.107 - 0.994i$
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  + (0.260 − 0.260i)2-s + (−0.826 − 1.52i)3-s + 1.86i·4-s + (0.895 + 2.04i)5-s + (−0.611 − 0.180i)6-s + (1.00 + 1.00i)8-s + (−1.63 + 2.51i)9-s + (0.766 + 0.300i)10-s − 3.38i·11-s + (2.83 − 1.54i)12-s + (−1.59 + 1.59i)13-s + (2.37 − 3.05i)15-s − 3.20·16-s + (−0.140 + 0.140i)17-s + (0.230 + 1.07i)18-s + 7.34i·19-s + ⋯
L(s)  = 1  + (0.184 − 0.184i)2-s + (−0.477 − 0.878i)3-s + 0.932i·4-s + (0.400 + 0.916i)5-s + (−0.249 − 0.0738i)6-s + (0.355 + 0.355i)8-s + (−0.544 + 0.838i)9-s + (0.242 + 0.0949i)10-s − 1.02i·11-s + (0.819 − 0.445i)12-s + (−0.442 + 0.442i)13-s + (0.614 − 0.789i)15-s − 0.801·16-s + (−0.0341 + 0.0341i)17-s + (0.0542 + 0.254i)18-s + 1.68i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.107 - 0.994i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.107 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866061 + 0.777551i\)
\(L(\frac12)\) \(\approx\) \(0.866061 + 0.777551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.826 + 1.52i)T \)
5 \( 1 + (-0.895 - 2.04i)T \)
7 \( 1 \)
good2 \( 1 + (-0.260 + 0.260i)T - 2iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (1.59 - 1.59i)T - 13iT^{2} \)
17 \( 1 + (0.140 - 0.140i)T - 17iT^{2} \)
19 \( 1 - 7.34iT - 19T^{2} \)
23 \( 1 + (-2.21 - 2.21i)T + 23iT^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 + 0.922T + 31T^{2} \)
37 \( 1 + (-5.91 - 5.91i)T + 37iT^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 + 0.864i)T - 43iT^{2} \)
47 \( 1 + (0.651 - 0.651i)T - 47iT^{2} \)
53 \( 1 + (-6.54 - 6.54i)T + 53iT^{2} \)
59 \( 1 - 6.25T + 59T^{2} \)
61 \( 1 + 1.83T + 61T^{2} \)
67 \( 1 + (0.815 + 0.815i)T + 67iT^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (-4.80 + 4.80i)T - 73iT^{2} \)
79 \( 1 + 3.41iT - 79T^{2} \)
83 \( 1 + (-6.26 - 6.26i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-6.71 - 6.71i)T + 97iT^{2} \)
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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

−10.97164375516819221401067802095, −9.846104923400789949997651023880, −8.669614809485619667548392612586, −7.74689309055958238329445021977, −7.19722485938140383020780731930, −6.19492760419646791592973776688, −5.45902071443195666714302266589, −3.88429094504634578212730018207, −2.90791828933063424317652985260, −1.81959491896466135887364438896, 0.58449762886151323203508583119, 2.26743753197604563042716142804, 4.12933941550279734197699894518, 4.95088390522226867282025201395, 5.37816221234523415271565920669, 6.40824202723865154488394940965, 7.40160741109045659993089393747, 8.947471532964935552965213041114, 9.406252606518725296981258949000, 10.06155458483314090793617680004

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