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Properties

Label 2-735-105.89-c1-0-52
Degree $2$
Conductor $735$
Sign $-0.580 + 0.814i$
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  + (1.22 − 2.12i)2-s + (−1.5 + 0.866i)3-s + (−2.01 − 3.49i)4-s + (1.93 + 1.11i)5-s + 4.25i·6-s − 4.98·8-s + (1.5 − 2.59i)9-s + (4.75 − 2.74i)10-s + (6.04 + 3.49i)12-s − 3.87·15-s + (−2.09 + 3.62i)16-s + (6.98 − 4.03i)17-s + (−3.68 − 6.38i)18-s + (−4.87 − 2.81i)19-s − 9.01i·20-s + ⋯
L(s)  = 1  + (0.868 − 1.50i)2-s + (−0.866 + 0.499i)3-s + (−1.00 − 1.74i)4-s + (0.866 + 0.499i)5-s + 1.73i·6-s − 1.76·8-s + (0.5 − 0.866i)9-s + (1.50 − 0.868i)10-s + (1.74 + 1.00i)12-s − 1.00·15-s + (−0.523 + 0.907i)16-s + (1.69 − 0.977i)17-s + (−0.868 − 1.50i)18-s + (−1.11 − 0.645i)19-s − 2.01i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.580 + 0.814i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908986 - 1.76453i\)
\(L(\frac12)\) \(\approx\) \(0.908986 - 1.76453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 \)
good2 \( 1 + (-1.22 + 2.12i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-6.98 + 4.03i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.87 + 2.81i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.79 + 8.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (3.83 - 2.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-0.886 - 0.511i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.71 - 8.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.53 + 2.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.91 + 5.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.54969362307063072082102444035, −9.701429556246009728346093741217, −8.993672114981928276659238424352, −7.09513752469574518911916901465, −6.11074042885793722850563468179, −5.25457323704571158880267260437, −4.59454772333010287813250008586, −3.38170531237160392159990458876, −2.47235780953709382135403999091, −0.966597498509929579732482621957, 1.59381485743390049544816942015, 3.65949021584265358898443249810, 4.83079270396765441386165308547, 5.70999386846098580632322951646, 5.90511688251882993516150809223, 6.97642298077716148281432396715, 7.76127202010140399448932022227, 8.556283808726046721536824026812, 9.755869848860232466853927010991, 10.66487165883362775612501909145

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