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Properties

Label 2-735-21.20-c1-0-27
Degree $2$
Conductor $735$
Sign $0.963 - 0.267i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.963 - 0.267i$
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: \((2,\ 735,\ (\ :1/2),\ 0.963 - 0.267i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66506 + 0.362691i\)
\(L(\frac12)\) \(\approx\) \(2.66506 + 0.362691i\)
\(L(\frac{3}{2})\) 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

−10.29571954089768530426858876493, −9.659026691546758505104128759557, −8.872374223672475488262652794734, −7.83928139832387695829710984634, −6.95039726079343459389425159605, −6.19593762351343304343250163525, −4.69174910463677022044188097278, −3.85895862231638651217171724676, −2.41935722358526563654089335616, −1.97044911053977487771165073377, 1.48045340051802795706594012770, 2.64298805089011233249058272186, 3.43512002161642410788794059295, 5.08149338862684713160346306139, 6.19787751797674601298504250804, 6.87348797025503953247056631989, 7.69320284265363860451378096504, 8.609318422628277716293956965055, 9.203996305859508553591928424639, 10.50942763221436089528682128684

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