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Properties

Label 2-735-21.20-c1-0-26
Degree $2$
Conductor $735$
Sign $0.831 + 0.555i$
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  + 2.33i·2-s + (−0.459 − 1.66i)3-s − 3.43·4-s − 5-s + (3.89 − 1.07i)6-s − 3.33i·8-s + (−2.57 + 1.53i)9-s − 2.33i·10-s + 2.79i·11-s + (1.57 + 5.73i)12-s − 3.20i·13-s + (0.459 + 1.66i)15-s + 0.919·16-s + 0.881·17-s + (−3.57 − 6.00i)18-s − 2.19i·19-s + ⋯
L(s)  = 1  + 1.64i·2-s + (−0.265 − 0.964i)3-s − 1.71·4-s − 0.447·5-s + (1.58 − 0.437i)6-s − 1.18i·8-s + (−0.859 + 0.511i)9-s − 0.737i·10-s + 0.842i·11-s + (0.455 + 1.65i)12-s − 0.888i·13-s + (0.118 + 0.431i)15-s + 0.229·16-s + 0.213·17-s + (−0.843 − 1.41i)18-s − 0.503i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.831 + 0.555i$
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: \((2,\ 735,\ (\ :1/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613073 - 0.185800i\)
\(L(\frac12)\) \(\approx\) \(0.613073 - 0.185800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.459 + 1.66i)T \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 - 2.33iT - 2T^{2} \)
11 \( 1 - 2.79iT - 11T^{2} \)
13 \( 1 + 3.20iT - 13T^{2} \)
17 \( 1 - 0.881T + 17T^{2} \)
19 \( 1 + 2.19iT - 19T^{2} \)
23 \( 1 + 7.54iT - 23T^{2} \)
29 \( 1 + 8.15iT - 29T^{2} \)
31 \( 1 + 8.80iT - 31T^{2} \)
37 \( 1 - 0.407T + 37T^{2} \)
41 \( 1 + 8.55T + 41T^{2} \)
43 \( 1 + 0.118T + 43T^{2} \)
47 \( 1 - 2.62T + 47T^{2} \)
53 \( 1 - 7.46iT - 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 + 12.3iT - 61T^{2} \)
67 \( 1 + 1.60T + 67T^{2} \)
71 \( 1 - 6.25iT - 71T^{2} \)
73 \( 1 - 0.221iT - 73T^{2} \)
79 \( 1 + 3.13T + 79T^{2} \)
83 \( 1 + 0.666T + 83T^{2} \)
89 \( 1 - 0.874T + 89T^{2} \)
97 \( 1 - 6.37iT - 97T^{2} \)
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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

−10.13013447974330686646825326653, −8.996659650166366040210709047773, −8.016863469306106293631842127289, −7.73570977412485726131144619382, −6.79167660597354074788266337060, −6.15753605595615609295256390502, −5.21499686540221504474418167305, −4.30042583102037069350827640259, −2.48534504275152927857890401738, −0.35773009837997953341171154808, 1.45332513618081570587672030460, 3.18468130546849626239897016283, 3.60258282790219538623126907440, 4.66216730826967437833835544870, 5.55506990697514139136869540075, 6.94822565984907001656080968620, 8.490038363622672519033852859667, 9.034946312044970037459035902976, 9.888997502696209111542076562595, 10.61197829416064299922674265491

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