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Properties

Label 2-735-105.23-c1-0-63
Degree $2$
Conductor $735$
Sign $0.367 + 0.929i$
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  + (2.17 − 0.582i)2-s + (0.245 − 1.71i)3-s + (2.64 − 1.52i)4-s + (1.39 + 1.74i)5-s + (−0.465 − 3.86i)6-s + (1.68 − 1.68i)8-s + (−2.87 − 0.840i)9-s + (4.05 + 2.97i)10-s + (3.88 − 2.24i)11-s + (−1.97 − 4.91i)12-s + (1.08 + 1.08i)13-s + (3.33 − 1.96i)15-s + (−0.381 + 0.660i)16-s + (0.548 − 2.04i)17-s + (−6.74 − 0.150i)18-s + (−3.66 − 2.11i)19-s + ⋯
L(s)  = 1  + (1.53 − 0.411i)2-s + (0.141 − 0.989i)3-s + (1.32 − 0.764i)4-s + (0.625 + 0.780i)5-s + (−0.189 − 1.57i)6-s + (0.595 − 0.595i)8-s + (−0.959 − 0.280i)9-s + (1.28 + 0.941i)10-s + (1.17 − 0.676i)11-s + (−0.569 − 1.41i)12-s + (0.300 + 0.300i)13-s + (0.861 − 0.508i)15-s + (−0.0952 + 0.165i)16-s + (0.133 − 0.496i)17-s + (−1.58 − 0.0354i)18-s + (−0.839 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.367 + 0.929i$
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: \((2,\ 735,\ (\ :1/2),\ 0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.28598 - 2.23432i\)
\(L(\frac12)\) \(\approx\) \(3.28598 - 2.23432i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.245 + 1.71i)T \)
5 \( 1 + (-1.39 - 1.74i)T \)
7 \( 1 \)
good2 \( 1 + (-2.17 + 0.582i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-3.88 + 2.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.08 - 1.08i)T + 13iT^{2} \)
17 \( 1 + (-0.548 + 2.04i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.66 + 2.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.840 + 3.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.69T + 29T^{2} \)
31 \( 1 + (0.530 + 0.918i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.54 - 5.75i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (-2.00 - 2.00i)T + 43iT^{2} \)
47 \( 1 + (5.10 - 1.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.34 + 2.23i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.35 - 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.88 - 6.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.569 - 0.152i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (1.13 - 4.22i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.78 + 3.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.0 + 11.0i)T - 83iT^{2} \)
89 \( 1 + (-1.75 + 3.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.60 + 5.60i)T - 97iT^{2} \)
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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

−10.69808359049734446957920525165, −9.377662209475830159281103198733, −8.467905114367139122598753378244, −7.12334090798367895420035141600, −6.25270343847214393718718991844, −6.09380147855125912918434143016, −4.68545774016652823511787093865, −3.46488448183282326226981674741, −2.67963353858429119564190979919, −1.59577482123937262732673871342, 2.03175636625058755444646178141, 3.59720679884090270328136033308, 4.17181100563512215722025679535, 5.05693335604945089715460614502, 5.82523797019507506252238737597, 6.52673230632403357438970190471, 7.899635282298861995513118904519, 9.002247969437194048416395861398, 9.599151769840974478667069141142, 10.60391811196375739396204114341

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