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Properties

Label 2-735-15.2-c1-0-40
Degree $2$
Conductor $735$
Sign $0.999 - 0.00590i$
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  + (−0.664 + 0.664i)2-s + (−0.578 + 1.63i)3-s + 1.11i·4-s + (0.459 − 2.18i)5-s + (−0.700 − 1.46i)6-s + (−2.07 − 2.07i)8-s + (−2.33 − 1.88i)9-s + (1.14 + 1.75i)10-s − 0.727i·11-s + (−1.82 − 0.646i)12-s + (1.44 − 1.44i)13-s + (3.30 + 2.01i)15-s + 0.515·16-s + (5.19 − 5.19i)17-s + (2.80 − 0.293i)18-s − 0.767i·19-s + ⋯
L(s)  = 1  + (−0.469 + 0.469i)2-s + (−0.334 + 0.942i)3-s + 0.558i·4-s + (0.205 − 0.978i)5-s + (−0.285 − 0.599i)6-s + (−0.732 − 0.732i)8-s + (−0.776 − 0.629i)9-s + (0.363 + 0.556i)10-s − 0.219i·11-s + (−0.526 − 0.186i)12-s + (0.400 − 0.400i)13-s + (0.853 + 0.520i)15-s + 0.128·16-s + (1.25 − 1.25i)17-s + (0.660 − 0.0691i)18-s − 0.175i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.999 - 0.00590i$
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: \((2,\ 735,\ (\ :1/2),\ 0.999 - 0.00590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881642 + 0.00260459i\)
\(L(\frac12)\) \(\approx\) \(0.881642 + 0.00260459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.578 - 1.63i)T \)
5 \( 1 + (-0.459 + 2.18i)T \)
7 \( 1 \)
good2 \( 1 + (0.664 - 0.664i)T - 2iT^{2} \)
11 \( 1 + 0.727iT - 11T^{2} \)
13 \( 1 + (-1.44 + 1.44i)T - 13iT^{2} \)
17 \( 1 + (-5.19 + 5.19i)T - 17iT^{2} \)
19 \( 1 + 0.767iT - 19T^{2} \)
23 \( 1 + (2.29 + 2.29i)T + 23iT^{2} \)
29 \( 1 - 4.07T + 29T^{2} \)
31 \( 1 + 0.419T + 31T^{2} \)
37 \( 1 + (4.45 + 4.45i)T + 37iT^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 - 5.15i)T - 43iT^{2} \)
47 \( 1 + (-4.97 + 4.97i)T - 47iT^{2} \)
53 \( 1 + (3.85 + 3.85i)T + 53iT^{2} \)
59 \( 1 - 1.61T + 59T^{2} \)
61 \( 1 - 9.57T + 61T^{2} \)
67 \( 1 + (-5.05 - 5.05i)T + 67iT^{2} \)
71 \( 1 - 7.06iT - 71T^{2} \)
73 \( 1 + (-11.1 + 11.1i)T - 73iT^{2} \)
79 \( 1 - 6.70iT - 79T^{2} \)
83 \( 1 + (1.83 + 1.83i)T + 83iT^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (5.62 + 5.62i)T + 97iT^{2} \)
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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

−9.961683287012035904254893813750, −9.575815909227708254650131823254, −8.571209207232042153511150579624, −8.159860577490625840600817577768, −6.92174792972608916696310421734, −5.81881408445496225308711762384, −5.05269481944408844254445989420, −3.97702003780769688362611589748, −2.97526780272358179873521549495, −0.60432524477342319870834638768, 1.34780746259208495613175430882, 2.25652110320311666055317311569, 3.52647938042362197515062152695, 5.34353680711599960558769090366, 6.08533921618220804006712090462, 6.76103630835881218777834510985, 7.82395196600430846829943627212, 8.628205999648206793412677667409, 9.838406601215134034315625993168, 10.42194039461749663042030528161

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