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Properties

Label 2-735-105.32-c1-0-38
Degree $2$
Conductor $735$
Sign $-0.856 + 0.516i$
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  + (−0.355 − 0.0952i)2-s + (−1.73 − 0.0448i)3-s + (−1.61 − 0.932i)4-s + (−1.32 + 1.79i)5-s + (0.611 + 0.180i)6-s + (1.00 + 1.00i)8-s + (2.99 + 0.155i)9-s + (0.643 − 0.513i)10-s + (2.93 + 1.69i)11-s + (2.75 + 1.68i)12-s + (1.59 − 1.59i)13-s + (2.37 − 3.05i)15-s + (1.60 + 2.77i)16-s + (0.0514 + 0.192i)17-s + (−1.05 − 0.340i)18-s + (−6.36 + 3.67i)19-s + ⋯
L(s)  = 1  + (−0.251 − 0.0673i)2-s + (−0.999 − 0.0258i)3-s + (−0.807 − 0.466i)4-s + (−0.593 + 0.804i)5-s + (0.249 + 0.0738i)6-s + (0.355 + 0.355i)8-s + (0.998 + 0.0517i)9-s + (0.203 − 0.162i)10-s + (0.884 + 0.510i)11-s + (0.795 + 0.486i)12-s + (0.442 − 0.442i)13-s + (0.614 − 0.789i)15-s + (0.400 + 0.694i)16-s + (0.0124 + 0.0466i)17-s + (−0.247 − 0.0802i)18-s + (−1.45 + 0.842i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0309522 - 0.111243i\)
\(L(\frac12)\) \(\approx\) \(0.0309522 - 0.111243i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0448i)T \)
5 \( 1 + (1.32 - 1.79i)T \)
7 \( 1 \)
good2 \( 1 + (0.355 + 0.0952i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-2.93 - 1.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.59 + 1.59i)T - 13iT^{2} \)
17 \( 1 + (-0.0514 - 0.192i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.36 - 3.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.810 + 3.02i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.49T + 29T^{2} \)
31 \( 1 + (0.461 - 0.798i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.16 + 8.08i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.39iT - 41T^{2} \)
43 \( 1 + (-0.864 + 0.864i)T - 43iT^{2} \)
47 \( 1 + (0.889 + 0.238i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.93 - 2.39i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (-1.11 + 0.298i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (1.75 + 6.56i)T + (-63.2 + 36.5i)T^{2} \)
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

−10.20666584848641611225672267041, −9.355929633100217114791734391948, −8.310168914103718689990887366100, −7.34833704081931660761899308798, −6.37871426574446457197875843839, −5.69009727274163964971337380493, −4.39672695176548311862093379777, −3.82905974180850112883249598840, −1.74790669868725049976926946246, −0.086533471695054021122153393305, 1.28228217602124544509196216094, 3.73481620437098511693919710784, 4.33001171854741860567545755368, 5.22747800136740147810887033798, 6.33550147578907606434303407859, 7.29964941380890334058549157188, 8.284760651074885614477282970185, 9.058152201477220814745345497570, 9.635648164963574831238211665992, 10.97840294317149844952642799600

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