Properties

Label 2-735-15.2-c1-0-57
Degree $2$
Conductor $735$
Sign $0.634 + 0.773i$
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  + (0.664 − 0.664i)2-s + (1.63 − 0.578i)3-s + 1.11i·4-s + (0.459 − 2.18i)5-s + (0.700 − 1.46i)6-s + (2.07 + 2.07i)8-s + (2.33 − 1.88i)9-s + (−1.14 − 1.75i)10-s + 0.727i·11-s + (0.646 + 1.82i)12-s + (−1.44 + 1.44i)13-s + (−0.515 − 3.83i)15-s + 0.515·16-s + (5.19 − 5.19i)17-s + (0.293 − 2.80i)18-s + 0.767i·19-s + ⋯
L(s)  = 1  + (0.469 − 0.469i)2-s + (0.942 − 0.334i)3-s + 0.558i·4-s + (0.205 − 0.978i)5-s + (0.285 − 0.599i)6-s + (0.732 + 0.732i)8-s + (0.776 − 0.629i)9-s + (−0.363 − 0.556i)10-s + 0.219i·11-s + (0.186 + 0.526i)12-s + (−0.400 + 0.400i)13-s + (−0.133 − 0.991i)15-s + 0.128·16-s + (1.25 − 1.25i)17-s + (0.0691 − 0.660i)18-s + 0.175i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.634 + 0.773i$
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: \((2,\ 735,\ (\ :1/2),\ 0.634 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58752 - 1.22413i\)
\(L(\frac12)\) \(\approx\) \(2.58752 - 1.22413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 + 0.578i)T \)
5 \( 1 + (-0.459 + 2.18i)T \)
7 \( 1 \)
good2 \( 1 + (-0.664 + 0.664i)T - 2iT^{2} \)
11 \( 1 - 0.727iT - 11T^{2} \)
13 \( 1 + (1.44 - 1.44i)T - 13iT^{2} \)
17 \( 1 + (-5.19 + 5.19i)T - 17iT^{2} \)
19 \( 1 - 0.767iT - 19T^{2} \)
23 \( 1 + (-2.29 - 2.29i)T + 23iT^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 - 0.419T + 31T^{2} \)
37 \( 1 + (4.45 + 4.45i)T + 37iT^{2} \)
41 \( 1 - 4.44iT - 41T^{2} \)
43 \( 1 + (5.15 - 5.15i)T - 43iT^{2} \)
47 \( 1 + (-4.97 + 4.97i)T - 47iT^{2} \)
53 \( 1 + (-3.85 - 3.85i)T + 53iT^{2} \)
59 \( 1 - 1.61T + 59T^{2} \)
61 \( 1 + 9.57T + 61T^{2} \)
67 \( 1 + (-5.05 - 5.05i)T + 67iT^{2} \)
71 \( 1 + 7.06iT - 71T^{2} \)
73 \( 1 + (11.1 - 11.1i)T - 73iT^{2} \)
79 \( 1 - 6.70iT - 79T^{2} \)
83 \( 1 + (1.83 + 1.83i)T + 83iT^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \)
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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

−10.03211189868178768905774824569, −9.335242022997687818943784975145, −8.580589847273714236608530192441, −7.68078039733278760877619036309, −7.13365951105109447909103539168, −5.48224390573806216737916671233, −4.57993356332144637607605883815, −3.61351098797431580746355261257, −2.60981636861754704071151764954, −1.45285797262224882716125973356, 1.74770689437084157650291853134, 3.08522018851300574621899860915, 3.95623454297946745081491373742, 5.17990249381614560963710770811, 6.03517787774143039857338374332, 7.04529364156974005761340892863, 7.72029880267550769142015632215, 8.778158328329351060298796922854, 9.883204134056032940130038863161, 10.31127340038399318820090690131

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