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Properties

Label 2-735-105.23-c1-0-5
Degree $2$
Conductor $735$
Sign $-0.136 - 0.990i$
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  + (−2.17 + 0.582i)2-s + (0.644 − 1.60i)3-s + (2.64 − 1.52i)4-s + (−1.39 − 1.74i)5-s + (−0.465 + 3.86i)6-s + (−1.68 + 1.68i)8-s + (−2.16 − 2.07i)9-s + (4.05 + 2.97i)10-s + (−3.88 + 2.24i)11-s + (−0.750 − 5.24i)12-s + (1.08 + 1.08i)13-s + (−3.70 + 1.12i)15-s + (−0.381 + 0.660i)16-s + (−0.548 + 2.04i)17-s + (5.91 + 3.24i)18-s + (−3.66 − 2.11i)19-s + ⋯
L(s)  = 1  + (−1.53 + 0.411i)2-s + (0.372 − 0.928i)3-s + (1.32 − 0.764i)4-s + (−0.625 − 0.780i)5-s + (−0.189 + 1.57i)6-s + (−0.595 + 0.595i)8-s + (−0.722 − 0.691i)9-s + (1.28 + 0.941i)10-s + (−1.17 + 0.676i)11-s + (−0.216 − 1.51i)12-s + (0.300 + 0.300i)13-s + (−0.957 + 0.289i)15-s + (−0.0952 + 0.165i)16-s + (−0.133 + 0.496i)17-s + (1.39 + 0.764i)18-s + (−0.839 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.136 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120537 + 0.138297i\)
\(L(\frac12)\) \(\approx\) \(0.120537 + 0.138297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.644 + 1.60i)T \)
5 \( 1 + (1.39 + 1.74i)T \)
7 \( 1 \)
good2 \( 1 + (2.17 - 0.582i)T + (1.73 - i)T^{2} \)
11 \( 1 + (3.88 - 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.08 - 1.08i)T + 13iT^{2} \)
17 \( 1 + (0.548 - 2.04i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.66 + 2.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.840 - 3.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + (0.530 + 0.918i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 5.75i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 - 2.00i)T + 43iT^{2} \)
47 \( 1 + (-5.10 + 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.34 - 2.23i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.35 + 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.88 - 6.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.569 - 0.152i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (1.13 - 4.22i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.78 + 3.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.0 - 11.0i)T - 83iT^{2} \)
89 \( 1 + (1.75 - 3.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.60 + 5.60i)T - 97iT^{2} \)
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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

−10.42474970661075855825779163886, −9.391813179451563500708423351415, −8.679927707490378130961715020785, −8.129339275996112770747213709927, −7.44219889942080467327014891012, −6.75655074450281651733958045589, −5.57840710353602053988124179049, −4.16116711012809474131169295222, −2.42462802787659467346383588273, −1.24145560916168845192180066165, 0.15900952835101563588543143648, 2.43473866813060263716035955677, 3.13222584141544492540494049077, 4.40797104118921389408034258261, 5.76843364845955845753226581366, 7.10822766964396496952955578488, 8.056198565113633585294272503674, 8.411285790968846302577911209303, 9.342783643649380859573393841983, 10.31623446574094703276768051061

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