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Properties

Label 2-735-15.8-c1-0-28
Degree $2$
Conductor $735$
Sign $-0.999 - 0.0203i$
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  + (−1.59 − 1.59i)2-s + (−1.36 − 1.06i)3-s + 3.05i·4-s + (−0.812 + 2.08i)5-s + (0.465 + 3.86i)6-s + (1.68 − 1.68i)8-s + (0.711 + 2.91i)9-s + (4.60 − 2.02i)10-s + 4.48i·11-s + (3.27 − 4.16i)12-s + (−1.08 − 1.08i)13-s + (3.33 − 1.96i)15-s + 0.762·16-s + (−1.49 − 1.49i)17-s + (3.50 − 5.76i)18-s − 4.22i·19-s + ⋯
L(s)  = 1  + (−1.12 − 1.12i)2-s + (−0.786 − 0.617i)3-s + 1.52i·4-s + (−0.363 + 0.931i)5-s + (0.189 + 1.57i)6-s + (0.595 − 0.595i)8-s + (0.237 + 0.971i)9-s + (1.45 − 0.639i)10-s + 1.35i·11-s + (0.944 − 1.20i)12-s + (−0.300 − 0.300i)13-s + (0.861 − 0.508i)15-s + 0.190·16-s + (−0.363 − 0.363i)17-s + (0.825 − 1.35i)18-s − 0.969i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 - 0.0203i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.999 - 0.0203i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00215393 + 0.211738i\)
\(L(\frac12)\) \(\approx\) \(0.00215393 + 0.211738i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.36 + 1.06i)T \)
5 \( 1 + (0.812 - 2.08i)T \)
7 \( 1 \)
good2 \( 1 + (1.59 + 1.59i)T + 2iT^{2} \)
11 \( 1 - 4.48iT - 11T^{2} \)
13 \( 1 + (1.08 + 1.08i)T + 13iT^{2} \)
17 \( 1 + (1.49 + 1.49i)T + 17iT^{2} \)
19 \( 1 + 4.22iT - 19T^{2} \)
23 \( 1 + (2.29 - 2.29i)T - 23iT^{2} \)
29 \( 1 + 1.69T + 29T^{2} \)
31 \( 1 + 1.06T + 31T^{2} \)
37 \( 1 + (-4.21 + 4.21i)T - 37iT^{2} \)
41 \( 1 + 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 - 2.00i)T + 43iT^{2} \)
47 \( 1 + (3.73 + 3.73i)T + 47iT^{2} \)
53 \( 1 + (-6.11 + 6.11i)T - 53iT^{2} \)
59 \( 1 - 4.70T + 59T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 + (0.416 - 0.416i)T - 67iT^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (-3.08 - 3.08i)T + 73iT^{2} \)
79 \( 1 - 6.67iT - 79T^{2} \)
83 \( 1 + (11.0 - 11.0i)T - 83iT^{2} \)
89 \( 1 - 3.51T + 89T^{2} \)
97 \( 1 + (5.60 - 5.60i)T - 97iT^{2} \)
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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

−10.06976739040495633527954117218, −9.480548372972265734968501939881, −8.214795042183409250613042069323, −7.35152949244229481110918825915, −6.89803777883300967255135324838, −5.51081090726303624404016766935, −4.18033855138143348000948265210, −2.69796174177977367386802618177, −1.90842670895994159586221310581, −0.21133174552475848237652407966, 1.07293395285700690254401514289, 3.67070403139881924395689498282, 4.76707834331825568475519934513, 5.85482278057862467791895258157, 6.24672727031290064891597519621, 7.52031971572325244507058423836, 8.359325417387891698769747074851, 8.948361585605209635669357001281, 9.733850666573119166050823629923, 10.53234321587847045139934739251

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