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Properties

Label 2-735-105.104-c1-0-70
Degree $2$
Conductor $735$
Sign $0.156 + 0.987i$
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  + 2.45·2-s − 1.73i·3-s + 4.03·4-s − 2.23i·5-s − 4.25i·6-s + 4.98·8-s − 2.99·9-s − 5.49i·10-s − 6.98i·12-s − 3.87·15-s + 4.19·16-s + 8.06i·17-s − 7.36·18-s − 5.62i·19-s − 9.01i·20-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.999i·3-s + 2.01·4-s − 0.999i·5-s − 1.73i·6-s + 1.76·8-s − 0.999·9-s − 1.73i·10-s − 2.01i·12-s − 0.999·15-s + 1.04·16-s + 1.95i·17-s − 1.73·18-s − 1.29i·19-s − 2.01i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01520 - 2.57490i\)
\(L(\frac12)\) \(\approx\) \(3.01520 - 2.57490i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + 2.23iT \)
7 \( 1 \)
good2 \( 1 - 2.45T + 2T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 8.06iT - 17T^{2} \)
19 \( 1 + 5.62iT - 19T^{2} \)
23 \( 1 - 9.58T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4.42iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 1.02iT - 47T^{2} \)
53 \( 1 - 9.43T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 4.08iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 5.83T + 79T^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.72403285965587659356639900631, −9.055162184233632104065484299155, −8.342608783501796109676806698376, −7.17466154433091162502011989568, −6.48158104637942069324271231277, −5.53792746586298195211712858449, −4.88113362675234728932986694625, −3.76215833450480333689431179457, −2.59862615173252316792534236377, −1.35360987229339559928335704982, 2.58860331521543202201731583219, 3.20709784012133919765093998174, 4.13523522116466720489947671982, 5.08862613055917404078844906831, 5.78034352333377268780883799186, 6.78650340920078416595711722034, 7.56653485620405067707913036115, 9.075637560150727912544095930668, 10.01772450706269686019815765952, 10.86909383120797948597968176213

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