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Properties

Label 2-735-15.2-c1-0-30
Degree $2$
Conductor $735$
Sign $-0.863 - 0.504i$
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  + (−1.72 + 1.72i)2-s + (1.44 + 0.953i)3-s − 3.95i·4-s + (1.96 + 1.07i)5-s + (−4.13 + 0.849i)6-s + (3.36 + 3.36i)8-s + (1.18 + 2.75i)9-s + (−5.23 + 1.53i)10-s + 3.55i·11-s + (3.76 − 5.71i)12-s + (1.28 − 1.28i)13-s + (1.81 + 3.42i)15-s − 3.71·16-s + (2.16 − 2.16i)17-s + (−6.79 − 2.71i)18-s + 0.383i·19-s + ⋯
L(s)  = 1  + (−1.21 + 1.21i)2-s + (0.834 + 0.550i)3-s − 1.97i·4-s + (0.877 + 0.478i)5-s + (−1.68 + 0.346i)6-s + (1.19 + 1.19i)8-s + (0.393 + 0.919i)9-s + (−1.65 + 0.486i)10-s + 1.07i·11-s + (1.08 − 1.64i)12-s + (0.356 − 0.356i)13-s + (0.469 + 0.883i)15-s − 0.927·16-s + (0.524 − 0.524i)17-s + (−1.60 − 0.640i)18-s + 0.0878i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.330706 + 1.22128i\)
\(L(\frac12)\) \(\approx\) \(0.330706 + 1.22128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.44 - 0.953i)T \)
5 \( 1 + (-1.96 - 1.07i)T \)
7 \( 1 \)
good2 \( 1 + (1.72 - 1.72i)T - 2iT^{2} \)
11 \( 1 - 3.55iT - 11T^{2} \)
13 \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \)
17 \( 1 + (-2.16 + 2.16i)T - 17iT^{2} \)
19 \( 1 - 0.383iT - 19T^{2} \)
23 \( 1 + (1.79 + 1.79i)T + 23iT^{2} \)
29 \( 1 - 5.51T + 29T^{2} \)
31 \( 1 + 0.647T + 31T^{2} \)
37 \( 1 + (3.66 + 3.66i)T + 37iT^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (0.335 - 0.335i)T - 43iT^{2} \)
47 \( 1 + (2.05 - 2.05i)T - 47iT^{2} \)
53 \( 1 + (-2.22 - 2.22i)T + 53iT^{2} \)
59 \( 1 + 7.63T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + (9.07 + 9.07i)T + 67iT^{2} \)
71 \( 1 + 3.06iT - 71T^{2} \)
73 \( 1 + (2.32 - 2.32i)T - 73iT^{2} \)
79 \( 1 + 3.70iT - 79T^{2} \)
83 \( 1 + (0.973 + 0.973i)T + 83iT^{2} \)
89 \( 1 + 3.03T + 89T^{2} \)
97 \( 1 + (10.3 + 10.3i)T + 97iT^{2} \)
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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

−10.11504934049388856446086571365, −9.785452045885559002400754075478, −9.032150458530975456101690101822, −8.181053404618017347260809218774, −7.42229381978785314225034576635, −6.64571337338309621889200167297, −5.65468353014933522049060092921, −4.65682624392267946621966708088, −2.98697719366353075471553189731, −1.65342822130207684315142435876, 0.969643247136870830892598713814, 1.88205167295683520687360738868, 2.92725130167123899612955392666, 3.87992506131999847960412683978, 5.67721197051177801893465573208, 6.78226910893676001187658355524, 8.014099315939616501507748188412, 8.602765308834684094942699672156, 9.093832598557543457489412182553, 9.980684609957217260542013481761

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