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Properties

Label 2-735-21.5-c1-0-42
Degree $2$
Conductor $735$
Sign $0.665 - 0.745i$
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  + (2.01 + 1.16i)2-s + (1.67 − 0.436i)3-s + (1.71 + 2.97i)4-s + (0.5 − 0.866i)5-s + (3.89 + 1.07i)6-s + 3.33i·8-s + (2.61 − 1.46i)9-s + (2.01 − 1.16i)10-s + (−2.42 + 1.39i)11-s + (4.17 + 4.23i)12-s + 3.20i·13-s + (0.459 − 1.66i)15-s + (−0.459 + 0.795i)16-s + (−0.440 − 0.763i)17-s + (6.99 + 0.0942i)18-s + (−1.90 − 1.09i)19-s + ⋯
L(s)  = 1  + (1.42 + 0.824i)2-s + (0.967 − 0.252i)3-s + (0.858 + 1.48i)4-s + (0.223 − 0.387i)5-s + (1.58 + 0.437i)6-s + 1.18i·8-s + (0.872 − 0.488i)9-s + (0.638 − 0.368i)10-s + (−0.729 + 0.421i)11-s + (1.20 + 1.22i)12-s + 0.888i·13-s + (0.118 − 0.431i)15-s + (−0.114 + 0.198i)16-s + (−0.106 − 0.185i)17-s + (1.64 + 0.0222i)18-s + (−0.436 − 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.665 - 0.745i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (656, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.665 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.04067 + 1.80938i\)
\(L(\frac12)\) \(\approx\) \(4.04067 + 1.80938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.436i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (-2.01 - 1.16i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (2.42 - 1.39i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.20iT - 13T^{2} \)
17 \( 1 + (0.440 + 0.763i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.90 + 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.53 + 3.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + (-7.62 + 4.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.203 - 0.352i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.55T + 41T^{2} \)
43 \( 1 + 0.118T + 43T^{2} \)
47 \( 1 + (1.31 - 2.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.46 - 3.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.04 - 3.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.7 + 6.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.802 - 1.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.25iT - 71T^{2} \)
73 \( 1 + (0.192 - 0.110i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.56 + 2.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.666T + 83T^{2} \)
89 \( 1 + (0.437 - 0.757i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.37iT - 97T^{2} \)
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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

−10.40399252739602083705406094240, −9.479365052533441143229352126808, −8.466945460243289917181989646321, −7.73892314066849652498250504602, −6.82067254791716202600111618616, −6.16111455194092257296876364600, −4.83358716634110017413757230529, −4.32348140494250194963292490720, −3.13142318537084356222022375381, −2.03493543807245956625007408701, 1.88824938007862633489650170611, 2.83342705105645288635475615362, 3.55104627037190032498322660297, 4.52400443985587069233798996557, 5.52383026314555772403833376908, 6.40102705999912114688603256834, 7.82053060127916921281441745712, 8.432219778683921564895203810809, 10.02296558311505921967950571827, 10.18445926515458692997620080200

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