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Properties

Label 2-735-105.23-c1-0-29
Degree $2$
Conductor $735$
Sign $0.581 - 0.813i$
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  + (−1.26 + 0.340i)2-s + (1.59 + 0.664i)3-s + (−0.236 + 0.136i)4-s + (−2.23 − 0.155i)5-s + (−2.25 − 0.299i)6-s + (2.11 − 2.11i)8-s + (2.11 + 2.12i)9-s + (2.88 − 0.560i)10-s + (3.38 − 1.95i)11-s + (−0.468 + 0.0611i)12-s + (1.56 + 1.56i)13-s + (−3.46 − 1.73i)15-s + (−1.69 + 2.92i)16-s + (0.693 − 2.58i)17-s + (−3.41 − 1.97i)18-s + (−1.61 − 0.930i)19-s + ⋯
L(s)  = 1  + (−0.897 + 0.240i)2-s + (0.923 + 0.383i)3-s + (−0.118 + 0.0681i)4-s + (−0.997 − 0.0697i)5-s + (−0.921 − 0.122i)6-s + (0.746 − 0.746i)8-s + (0.705 + 0.708i)9-s + (0.912 − 0.177i)10-s + (1.01 − 0.588i)11-s + (−0.135 + 0.0176i)12-s + (0.434 + 0.434i)13-s + (−0.894 − 0.447i)15-s + (−0.422 + 0.731i)16-s + (0.168 − 0.627i)17-s + (−0.803 − 0.466i)18-s + (−0.369 − 0.213i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.964263 + 0.496181i\)
\(L(\frac12)\) \(\approx\) \(0.964263 + 0.496181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.59 - 0.664i)T \)
5 \( 1 + (2.23 + 0.155i)T \)
7 \( 1 \)
good2 \( 1 + (1.26 - 0.340i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-3.38 + 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.56 - 1.56i)T + 13iT^{2} \)
17 \( 1 + (-0.693 + 2.58i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.61 + 0.930i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.638 + 2.38i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.513T + 29T^{2} \)
31 \( 1 + (-4.29 - 7.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.77 - 6.60i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (-5.10 + 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.85 - 0.498i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.259 - 0.448i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.55 + 4.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.74 + 2.34i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (-0.749 + 2.79i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.37 + 2.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.16 + 9.16i)T - 83iT^{2} \)
89 \( 1 + (5.67 - 9.82i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.81 + 6.81i)T - 97iT^{2} \)
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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

−10.27563382043159701770008441503, −9.310452846093874750467347570669, −8.738018613758065127979369979712, −8.235789962064007491222690363563, −7.34752386035859951601742304831, −6.55497049392667082055240200448, −4.66599457306603187272091552499, −4.02785511314262291752873982708, −3.06018529858853154182742252628, −1.12589805874211618709148354196, 0.924483411680953201646321564453, 2.19417923546279305710682612793, 3.73113368568761308704380143904, 4.35207232473037161158830389302, 6.01904967651476373642345578108, 7.28850871117582154279489450277, 7.80890203452513452793945372967, 8.635562328264448711419580458965, 9.193249778309860231557734698652, 10.07422565815878851504324703409

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