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Properties

Label 2-735-105.59-c1-0-6
Degree $2$
Conductor $735$
Sign $0.797 + 0.603i$
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  + (−1.10 − 1.91i)2-s + (−1.38 + 1.04i)3-s + (−1.43 + 2.49i)4-s + (−2.22 − 0.255i)5-s + (3.52 + 1.48i)6-s + 1.94·8-s + (0.810 − 2.88i)9-s + (1.96 + 4.53i)10-s + (−3.30 − 1.90i)11-s + (−0.622 − 4.94i)12-s − 6.50·13-s + (3.33 − 1.97i)15-s + (0.732 + 1.26i)16-s + (2.54 + 1.47i)17-s + (−6.42 + 1.64i)18-s + (−1.76 + 1.01i)19-s + ⋯
L(s)  = 1  + (−0.781 − 1.35i)2-s + (−0.796 + 0.604i)3-s + (−0.719 + 1.24i)4-s + (−0.993 − 0.114i)5-s + (1.43 + 0.606i)6-s + 0.687·8-s + (0.270 − 0.962i)9-s + (0.621 + 1.43i)10-s + (−0.996 − 0.575i)11-s + (−0.179 − 1.42i)12-s − 1.80·13-s + (0.860 − 0.509i)15-s + (0.183 + 0.317i)16-s + (0.617 + 0.356i)17-s + (−1.51 + 0.386i)18-s + (−0.404 + 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.797 + 0.603i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.797 + 0.603i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289189 - 0.0970738i\)
\(L(\frac12)\) \(\approx\) \(0.289189 - 0.0970738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.38 - 1.04i)T \)
5 \( 1 + (2.22 + 0.255i)T \)
7 \( 1 \)
good2 \( 1 + (1.10 + 1.91i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3.30 + 1.90i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.50T + 13T^{2} \)
17 \( 1 + (-2.54 - 1.47i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.76 - 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.50 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.25iT - 29T^{2} \)
31 \( 1 + (-5.77 - 3.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.91 + 1.68i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.51T + 41T^{2} \)
43 \( 1 + 7.03iT - 43T^{2} \)
47 \( 1 + (-4.34 + 2.50i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.967 - 1.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.8 + 6.83i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.34 - 5.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 + (-1.65 + 2.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.92 - 5.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.66iT - 83T^{2} \)
89 \( 1 + (6.75 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.46T + 97T^{2} \)
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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

−10.28129618448427905566558015969, −9.936138758011543092565284732338, −8.760480673245069120803008108525, −8.060933354473876652696718778631, −6.99901858464522817885006214235, −5.54596487864892249880547160198, −4.58802730482168263628121281497, −3.57858517111664520712730483978, −2.55485200474944397392278528292, −0.63280970994829087904061401168, 0.41496849333041086863742347426, 2.59645819265375800404136052744, 4.65468304904873522189462576243, 5.24407327820315262362544839306, 6.38657122904140315148745566075, 7.21618181675760672024633387040, 7.72604643612377042900674756014, 8.163013074471608194934146616267, 9.645000788184025297967991238134, 10.16212838743685963097852901226

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