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Properties

Label 2-735-105.104-c1-0-32
Degree $2$
Conductor $735$
Sign $-0.703 + 0.710i$
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  − 0.903·2-s + (−1.72 − 0.188i)3-s − 1.18·4-s + (−1.94 − 1.10i)5-s + (1.55 + 0.169i)6-s + 2.87·8-s + (2.92 + 0.647i)9-s + (1.75 + 0.994i)10-s + 4.06i·11-s + (2.03 + 0.222i)12-s + 2.88·13-s + (3.14 + 2.26i)15-s − 0.230·16-s − 6.66i·17-s + (−2.64 − 0.585i)18-s + 5.70i·19-s + ⋯
L(s)  = 1  − 0.638·2-s + (−0.994 − 0.108i)3-s − 0.591·4-s + (−0.870 − 0.492i)5-s + (0.634 + 0.0693i)6-s + 1.01·8-s + (0.976 + 0.215i)9-s + (0.556 + 0.314i)10-s + 1.22i·11-s + (0.588 + 0.0643i)12-s + 0.799·13-s + (0.811 + 0.583i)15-s − 0.0575·16-s − 1.61i·17-s + (−0.623 − 0.137i)18-s + 1.30i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.703 + 0.710i$
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: \((2,\ 735,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0871693 - 0.209059i\)
\(L(\frac12)\) \(\approx\) \(0.0871693 - 0.209059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.188i)T \)
5 \( 1 + (1.94 + 1.10i)T \)
7 \( 1 \)
good2 \( 1 + 0.903T + 2T^{2} \)
11 \( 1 - 4.06iT - 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 + 6.66iT - 17T^{2} \)
19 \( 1 - 5.70iT - 19T^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 + 5.89iT - 29T^{2} \)
31 \( 1 + 1.87iT - 31T^{2} \)
37 \( 1 + 1.59iT - 37T^{2} \)
41 \( 1 + 9.12T + 41T^{2} \)
43 \( 1 - 7.53iT - 43T^{2} \)
47 \( 1 + 6.91iT - 47T^{2} \)
53 \( 1 + 1.51T + 53T^{2} \)
59 \( 1 - 0.991T + 59T^{2} \)
61 \( 1 + 6.16iT - 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 + 4.81iT - 71T^{2} \)
73 \( 1 + 0.560T + 73T^{2} \)
79 \( 1 - 3.79T + 79T^{2} \)
83 \( 1 - 4.00iT - 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−9.899483188203367278122550195231, −9.432220712641949133914559219047, −8.184673248670752877071009317139, −7.64896710068783092540902441754, −6.72539717752577331751062765530, −5.39001168344014584765670772212, −4.62574632478755968251863360318, −3.85792229336810973173497337591, −1.55529681025302767798536476457, −0.21377367688774253364373770088, 1.15302477084272243409401203233, 3.46745815050095795262486177545, 4.24700126508016915508314978222, 5.36984547663214240741050489946, 6.38244270346205557124929327797, 7.23576409709760425294093184951, 8.390839324761208523758033834206, 8.746458239235901687096857259035, 10.07823539486202793888034783982, 10.84167672084801699074079377706

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