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Properties

Label 2-735-21.20-c1-0-23
Degree $2$
Conductor $735$
Sign $0.402 + 0.915i$
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  − 0.385i·2-s + (−1.56 − 0.742i)3-s + 1.85·4-s − 5-s + (−0.286 + 0.603i)6-s − 1.48i·8-s + (1.89 + 2.32i)9-s + 0.385i·10-s + 2.54i·11-s + (−2.89 − 1.37i)12-s − 3.06i·13-s + (1.56 + 0.742i)15-s + 3.12·16-s + 6.46·17-s + (0.896 − 0.731i)18-s − 1.19i·19-s + ⋯
L(s)  = 1  − 0.272i·2-s + (−0.903 − 0.428i)3-s + 0.925·4-s − 0.447·5-s + (−0.116 + 0.246i)6-s − 0.525i·8-s + (0.632 + 0.774i)9-s + 0.121i·10-s + 0.766i·11-s + (−0.836 − 0.396i)12-s − 0.850i·13-s + (0.404 + 0.191i)15-s + 0.782·16-s + 1.56·17-s + (0.211 − 0.172i)18-s − 0.274i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13290 - 0.739760i\)
\(L(\frac12)\) \(\approx\) \(1.13290 - 0.739760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 + 0.742i)T \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 + 0.385iT - 2T^{2} \)
11 \( 1 - 2.54iT - 11T^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
17 \( 1 - 6.46T + 17T^{2} \)
19 \( 1 + 1.19iT - 19T^{2} \)
23 \( 1 + 3.05iT - 23T^{2} \)
29 \( 1 + 7.77iT - 29T^{2} \)
31 \( 1 - 6.87iT - 31T^{2} \)
37 \( 1 + 3.55T + 37T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
43 \( 1 - 5.46T + 43T^{2} \)
47 \( 1 - 3.22T + 47T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 - 3.96T + 59T^{2} \)
61 \( 1 + 9.34iT - 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 + 0.921iT - 71T^{2} \)
73 \( 1 + 0.296iT - 73T^{2} \)
79 \( 1 - 8.29T + 79T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 + 18.8T + 89T^{2} \)
97 \( 1 - 12.3iT - 97T^{2} \)
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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

−10.38572494308176067214898023899, −9.816908036694528494194002442289, −8.140917693769490846023583385264, −7.51329150323085776476193138522, −6.76196467430189411602942557590, −5.83913798745003420010018873770, −4.93086836526453459228625302597, −3.57250932542498925664074489060, −2.29198701921345819333882683551, −0.911551087514009465052328805520, 1.29472352109834372482724904151, 3.11926045497046356494059146732, 4.11646811407879605975492672904, 5.48820681934510618508184175030, 5.95413670150715506450894129636, 7.05011408266147255501924715181, 7.65793371464112641434847539844, 8.841527803979821062927734319645, 9.871072858262395440324160873763, 10.74970698408504607916244357600

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