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Properties

Label 2-735-15.2-c1-0-32
Degree $2$
Conductor $735$
Sign $0.525 - 0.850i$
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 + (1.37 + 1.05i)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 + (−0.287 + 0.374i)12-s + (−1.56 + 1.56i)13-s + (−3.46 + 1.73i)15-s + 3.38·16-s + (−1.89 + 1.89i)17-s + (3.41 + 1.96i)18-s + 1.86i·19-s + ⋯
L(s)  = 1  + (0.657 − 0.657i)2-s + (0.794 + 0.607i)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 + (−0.0828 + 0.108i)12-s + (−0.434 + 0.434i)13-s + (−0.894 + 0.447i)15-s + 0.845·16-s + (−0.459 + 0.459i)17-s + (0.805 + 0.462i)18-s + 0.426i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.525 - 0.850i$
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: \((2,\ 735,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23459 + 1.24579i\)
\(L(\frac12)\) \(\approx\) \(2.23459 + 1.24579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.37 - 1.05i)T \)
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

−10.63177539694259734413370041865, −9.964301114371486623824333887246, −8.640013948690084012350793416091, −8.126358817176873524875559500723, −7.18612704831670754960828445246, −5.94810169548209925988294962751, −4.56805092590657623196780858403, −3.87165063009692105326832169417, −3.04397866851767640179292279150, −2.24765964538661239105866727961, 1.04505696045521244032096609462, 2.48582615544296166671268080963, 4.09982236596499198835566804523, 4.68780553840157093996024530322, 5.77997683565277716514391513505, 6.88130618973433835167442309633, 7.51617600707536121377061832349, 8.249567764346732739668391341418, 9.437531240291444665025160000591, 9.827186373043226733324637159435

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