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Properties

Label 2-735-35.3-c1-0-2
Degree $2$
Conductor $735$
Sign $-0.204 - 0.978i$
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  + (0.228 + 0.0611i)2-s + (−0.258 − 0.965i)3-s + (−1.68 − 0.972i)4-s + (1.18 + 1.89i)5-s − 0.236i·6-s + (−0.658 − 0.658i)8-s + (−0.866 + 0.499i)9-s + (0.155 + 0.504i)10-s + (−1.99 + 3.45i)11-s + (−0.503 + 1.87i)12-s + (−0.500 + 0.500i)13-s + (1.52 − 1.63i)15-s + (1.83 + 3.17i)16-s + (−2.29 + 0.614i)17-s + (−0.228 + 0.0611i)18-s + (−3.60 − 6.25i)19-s + ⋯
L(s)  = 1  + (0.161 + 0.0432i)2-s + (−0.149 − 0.557i)3-s + (−0.841 − 0.486i)4-s + (0.532 + 0.846i)5-s − 0.0964i·6-s + (−0.232 − 0.232i)8-s + (−0.288 + 0.166i)9-s + (0.0492 + 0.159i)10-s + (−0.600 + 1.04i)11-s + (−0.145 + 0.542i)12-s + (−0.138 + 0.138i)13-s + (0.392 − 0.423i)15-s + (0.458 + 0.794i)16-s + (−0.556 + 0.148i)17-s + (−0.0537 + 0.0144i)18-s + (−0.828 − 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.204 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454756 + 0.559518i\)
\(L(\frac12)\) \(\approx\) \(0.454756 + 0.559518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-1.18 - 1.89i)T \)
7 \( 1 \)
good2 \( 1 + (-0.228 - 0.0611i)T + (1.73 + i)T^{2} \)
11 \( 1 + (1.99 - 3.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.500 - 0.500i)T - 13iT^{2} \)
17 \( 1 + (2.29 - 0.614i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.60 + 6.25i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 - 7.04i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (-4.27 - 2.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.399 + 0.106i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \)
47 \( 1 + (-0.111 + 0.417i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.37 - 1.97i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.05 + 5.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.15 - 3.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.345 - 1.28i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (0.506 + 1.88i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.48 - 4.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 + (-3.91 - 6.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.43 + 7.43i)T + 97iT^{2} \)
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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

−10.59210416437660280234017889690, −9.747833701680530950363992097296, −9.107490594847177330544834886991, −7.933811972907351313471936402327, −6.96485735818316747317072090149, −6.28493274387596965283076977104, −5.24901261896262107999124573529, −4.42464146699763977946099151433, −2.90050104633489717067596054733, −1.71056851791872527573965417141, 0.35589677643201921255165083199, 2.48361659766354746164795447027, 3.86750807093778786467687380580, 4.58537545772291000324623507790, 5.53081156637520000568854517257, 6.23503664924017338998929313899, 8.033186692534461320237035644981, 8.436744652538014191336790320567, 9.220609025984694240131765888893, 10.10603680068818609851938044694

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