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Properties

Label 2-735-105.53-c1-0-25
Degree $2$
Conductor $735$
Sign $-0.320 - 0.947i$
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  + (−0.0952 + 0.355i)2-s + (−0.0448 + 1.73i)3-s + (1.61 + 0.932i)4-s + (2.22 − 0.249i)5-s + (−0.611 − 0.180i)6-s + (−1.00 + 1.00i)8-s + (−2.99 − 0.155i)9-s + (−0.123 + 0.813i)10-s + (2.93 + 1.69i)11-s + (−1.68 + 2.75i)12-s + (−1.59 − 1.59i)13-s + (0.331 + 3.85i)15-s + (1.60 + 2.77i)16-s + (−0.192 + 0.0514i)17-s + (0.340 − 1.05i)18-s + (−6.36 + 3.67i)19-s + ⋯
L(s)  = 1  + (−0.0673 + 0.251i)2-s + (−0.0258 + 0.999i)3-s + (0.807 + 0.466i)4-s + (0.993 − 0.111i)5-s + (−0.249 − 0.0738i)6-s + (−0.355 + 0.355i)8-s + (−0.998 − 0.0517i)9-s + (−0.0389 + 0.257i)10-s + (0.884 + 0.510i)11-s + (−0.486 + 0.795i)12-s + (−0.442 − 0.442i)13-s + (0.0856 + 0.996i)15-s + (0.400 + 0.694i)16-s + (−0.0466 + 0.0124i)17-s + (0.0802 − 0.247i)18-s + (−1.45 + 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.320 - 0.947i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16406 + 1.62217i\)
\(L(\frac12)\) \(\approx\) \(1.16406 + 1.62217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0448 - 1.73i)T \)
5 \( 1 + (-2.22 + 0.249i)T \)
7 \( 1 \)
good2 \( 1 + (0.0952 - 0.355i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.59 + 1.59i)T + 13iT^{2} \)
17 \( 1 + (0.192 - 0.0514i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.36 - 3.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.02 - 0.810i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 9.49T + 29T^{2} \)
31 \( 1 + (-0.461 + 0.798i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.08 + 2.16i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 - 0.864i)T + 43iT^{2} \)
47 \( 1 + (-0.238 + 0.889i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.39 + 8.93i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.12 + 5.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.916 - 1.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.298 + 1.11i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (6.56 - 1.75i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.95 + 1.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.26 - 6.26i)T - 83iT^{2} \)
89 \( 1 + (6.18 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.71 + 6.71i)T - 97iT^{2} \)
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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

−10.39749426939052571645683137995, −9.980921068435604295061500851329, −8.866961127078749909494563809031, −8.322850977138273359210555920036, −6.92729398696074757171219597217, −6.26212956454940242366871982538, −5.33183502941896878597414217523, −4.24910675414417921096687331372, −3.06026669551002774599200206009, −1.96605578352001812509187584075, 1.10010013958310437344689872531, 2.15030019995045000065811153030, 2.97964509771670784171522883352, 4.86533471353038591169542700336, 6.08119103895480839423172249947, 6.55016636412439535197219629924, 7.13230132747530317743936417581, 8.592545731844615248207007221563, 9.176132934073808785163007621082, 10.34375050213256094495416238833

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