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Properties

Label 2-735-105.59-c1-0-18
Degree $2$
Conductor $735$
Sign $0.327 - 0.944i$
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  + (0.866 + 1.5i)2-s + (−1.72 + 0.158i)3-s + (−0.5 + 0.866i)4-s + (−2.09 − 0.792i)5-s + (−1.73 − 2.44i)6-s + 1.73·8-s + (2.94 − 0.548i)9-s + (−0.621 − 3.82i)10-s + (−2.44 − 1.41i)11-s + (0.724 − 1.57i)12-s + 4·13-s + (3.73 + 1.03i)15-s + (2.49 + 4.33i)16-s + (2.44 + 1.41i)17-s + (3.37 + 3.94i)18-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.995 + 0.0917i)3-s + (−0.250 + 0.433i)4-s + (−0.935 − 0.354i)5-s + (−0.707 − 0.999i)6-s + 0.612·8-s + (0.983 − 0.182i)9-s + (−0.196 − 1.20i)10-s + (−0.738 − 0.426i)11-s + (0.209 − 0.454i)12-s + 1.10·13-s + (0.963 + 0.267i)15-s + (0.624 + 1.08i)16-s + (0.594 + 0.342i)17-s + (0.795 + 0.930i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17787 + 0.838077i\)
\(L(\frac12)\) \(\approx\) \(1.17787 + 0.838077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.158i)T \)
5 \( 1 + (2.09 + 0.792i)T \)
7 \( 1 \)
good2 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-2.44 - 1.41i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + (-8.48 - 4.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 + (2.44 - 1.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.48 + 4.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.24 - 2.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-4 + 6.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.80087541603942072641273047111, −9.822071982376792559029713888072, −8.342469889715229552797977040573, −7.84288586002163858887025663551, −6.83024623610636308245353750671, −6.04744976888685477689476551460, −5.28380724070844544905237881033, −4.48255227919687108077629007726, −3.54039697137990367852771823310, −1.07237799576244200151087470923, 0.971451510419160897582969466562, 2.59361735484706029040392904479, 3.72913146452092540499999678467, 4.53459496108177362174631549715, 5.43376173127693472378183883440, 6.68853712738342248636863720746, 7.49031381825688643941517122667, 8.370413458425083540367565672058, 9.949740695287959150539681070032, 10.58723846318217754201551947760

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