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Properties

Label 2-735-105.104-c1-0-40
Degree $2$
Conductor $735$
Sign $0.533 + 0.845i$
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 + (−3.55 + 1.52i)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 + (−0.918 + 0.394i)15-s − 0.0575·16-s − 1.61i·17-s + (0.623 + 0.137i)18-s − 1.30i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.533 + 0.845i$
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: \((2,\ 735,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22093 - 0.673180i\)
\(L(\frac12)\) \(\approx\) \(1.22093 - 0.673180i\)
\(L(\frac{3}{2})\) 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

−10.14215259670028953302245313986, −9.486289922181187083616511300426, −8.812166034336102150520445074095, −7.32355678411049488845248228785, −6.49455637363522219523233914069, −5.50613996065507503794445161792, −4.91960982918596494997159263218, −4.21015215750301770779993251881, −2.46607569338126127032676182817, −0.78071212717355042395536443366, 1.34218883745080480095745909988, 3.29102675702782789172746880653, 4.09717437533671915007862505779, 5.36803652688994289251144299090, 6.04862296806615499257364187878, 6.32973802844020567242195445970, 7.956737709156017875204471224400, 8.940525222614105485322355651740, 9.808660809201083894672217998375, 10.76711702733806061373322544905

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