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Properties

Label 2-735-105.59-c1-0-58
Degree $2$
Conductor $735$
Sign $-0.803 + 0.595i$
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  + (0.306 + 1.70i)3-s + (1 − 1.73i)4-s + (−1.93 + 1.11i)5-s + (−2.81 + 1.04i)9-s + (−5.12 − 2.95i)11-s + (3.25 + 1.17i)12-s − 2.64·13-s + (−2.5 − 2.95i)15-s + (−1.99 − 3.46i)16-s + (−1.93 − 1.11i)17-s + 4.47i·20-s + (2.5 − 4.33i)25-s + (−2.64 − 4.47i)27-s + 5.91i·29-s + (3.47 − 9.64i)33-s + ⋯
L(s)  = 1  + (0.177 + 0.984i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.499i)5-s + (−0.937 + 0.348i)9-s + (−1.54 − 0.891i)11-s + (0.940 + 0.338i)12-s − 0.733·13-s + (−0.645 − 0.763i)15-s + (−0.499 − 0.866i)16-s + (−0.469 − 0.271i)17-s + 0.999i·20-s + (0.5 − 0.866i)25-s + (−0.509 − 0.860i)27-s + 1.09i·29-s + (0.604 − 1.67i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.803 + 0.595i$
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: \((2,\ 735,\ (\ :1/2),\ -0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0446230 - 0.135039i\)
\(L(\frac12)\) \(\approx\) \(0.0446230 - 0.135039i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.306 - 1.70i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
11 \( 1 + (5.12 + 2.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + (1.93 + 1.11i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.91iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (9.68 - 5.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-5.29 + 9.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18.5T + 97T^{2} \)
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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

−10.25576850371377866625268741556, −9.368380969914482382456523676034, −8.302766408860814553068562162393, −7.53032192361578198664318962599, −6.44150101718114467927977264428, −5.33646793979217160602002521253, −4.71468423707253193836834271767, −3.27903497850911129168240966286, −2.53000351921565958319880799780, −0.06376460010592685334312796048, 2.06705041109976935303399422597, 2.94492298740847056535524040493, 4.24060394100272543928750123009, 5.33188276100516328644173972485, 6.72813367137609255600527696043, 7.42290806189520344017294218687, 7.994140838189322223460339747997, 8.554238566861824412403679581572, 9.826117936275046439169283269037, 11.02376377795217713132717321566

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