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Properties

Label 2-735-49.22-c1-0-4
Degree $2$
Conductor $735$
Sign $-0.880 - 0.473i$
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  + (0.991 − 0.477i)2-s + (−0.222 + 0.974i)3-s + (−0.491 + 0.616i)4-s + (−0.222 + 0.974i)5-s + (0.244 + 1.07i)6-s + (−1.66 − 2.05i)7-s + (−0.682 + 2.99i)8-s + (−0.900 − 0.433i)9-s + (0.244 + 1.07i)10-s + (1.44 − 0.696i)11-s + (−0.491 − 0.616i)12-s + (−4.16 + 2.00i)13-s + (−2.63 − 1.23i)14-s + (−0.900 − 0.433i)15-s + (0.400 + 1.75i)16-s + (1.82 + 2.28i)17-s + ⋯
L(s)  = 1  + (0.701 − 0.337i)2-s + (−0.128 + 0.562i)3-s + (−0.245 + 0.308i)4-s + (−0.0995 + 0.436i)5-s + (0.0999 + 0.438i)6-s + (−0.631 − 0.775i)7-s + (−0.241 + 1.05i)8-s + (−0.300 − 0.144i)9-s + (0.0774 + 0.339i)10-s + (0.435 − 0.209i)11-s + (−0.141 − 0.178i)12-s + (−1.15 + 0.556i)13-s + (−0.704 − 0.330i)14-s + (−0.232 − 0.112i)15-s + (0.100 + 0.438i)16-s + (0.441 + 0.554i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(5.86900\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{735} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 735,\ (\ :1/2),\ -0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197650 + 0.785059i\)
\(L(\frac12)\) \(\approx\) \(0.197650 + 0.785059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (1.66 + 2.05i)T \)
good2 \( 1 + (-0.991 + 0.477i)T + (1.24 - 1.56i)T^{2} \)
11 \( 1 + (-1.44 + 0.696i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (4.16 - 2.00i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-1.82 - 2.28i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + 3.21T + 19T^{2} \)
23 \( 1 + (3.76 - 4.72i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (2.93 + 3.68i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + 8.97T + 31T^{2} \)
37 \( 1 + (-5.34 - 6.69i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 + (-1.41 + 6.20i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (0.471 + 2.06i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (4.37 - 2.10i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.69 + 2.12i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-1.83 - 8.05i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (-1.27 - 1.60i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + (3.55 - 4.46i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-13.3 - 6.42i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + (-3.65 - 1.76i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (4.77 + 2.30i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + 3.55T + 97T^{2} \)
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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

−10.87314001977976466611537819622, −9.892814240662213415441259520878, −9.313261398876373775472143405944, −8.113588938143048349259408416017, −7.23737217873728916735989976518, −6.16918893872649989636318098355, −5.15135586400547945304024512545, −3.93508328240834357184428958710, −3.71969122310073652312972329340, −2.32766914270782846278596804384, 0.31875035698296767602586071668, 2.18222025485305658700109295578, 3.56882238336160611824868964835, 4.79376278899184797471361619629, 5.52277681345496338773359377019, 6.32653180656338237949972466381, 7.16095766189503127191717892528, 8.221023381254401327551362121716, 9.339494912399552231736740745991, 9.733356275682150103350689692819

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