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Properties

Label 2-210-15.8-c1-0-5
Degree $2$
Conductor $210$
Sign $0.356 - 0.934i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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  + (0.707 + 0.707i)2-s + (1.65 + 0.510i)3-s + 1.00i·4-s + (−1.97 + 1.04i)5-s + (0.809 + 1.53i)6-s + (0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (2.47 + 1.68i)9-s + (−2.13 − 0.655i)10-s − 0.598i·11-s + (−0.510 + 1.65i)12-s + (2.55 + 2.55i)13-s + 1.00·14-s + (−3.80 + 0.727i)15-s − 1.00·16-s + (−4.20 − 4.20i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.955 + 0.294i)3-s + 0.500i·4-s + (−0.883 + 0.468i)5-s + (0.330 + 0.625i)6-s + (0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.826 + 0.563i)9-s + (−0.676 − 0.207i)10-s − 0.180i·11-s + (−0.147 + 0.477i)12-s + (0.709 + 0.709i)13-s + 0.267·14-s + (−0.982 + 0.187i)15-s − 0.250·16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50854 + 1.03895i\)
\(L(\frac12)\) \(\approx\) \(1.50854 + 1.03895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-1.65 - 0.510i)T \)
5 \( 1 + (1.97 - 1.04i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + 0.598iT - 11T^{2} \)
13 \( 1 + (-2.55 - 2.55i)T + 13iT^{2} \)
17 \( 1 + (4.20 + 4.20i)T + 17iT^{2} \)
19 \( 1 + 5.70iT - 19T^{2} \)
23 \( 1 + (-2.23 + 2.23i)T - 23iT^{2} \)
29 \( 1 - 0.0410T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 + (-1.56 + 1.56i)T - 37iT^{2} \)
41 \( 1 - 5.79iT - 41T^{2} \)
43 \( 1 + (-0.325 - 0.325i)T + 43iT^{2} \)
47 \( 1 + (-1.56 - 1.56i)T + 47iT^{2} \)
53 \( 1 + (-2.01 + 2.01i)T - 53iT^{2} \)
59 \( 1 - 9.35T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + (-5.89 + 5.89i)T - 67iT^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (9.67 + 9.67i)T + 73iT^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + (-1.04 + 1.04i)T - 83iT^{2} \)
89 \( 1 + 18.1T + 89T^{2} \)
97 \( 1 + (4.69 - 4.69i)T - 97iT^{2} \)
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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

−12.89497423826833753277298006332, −11.43119502028042598905038032628, −10.90289277605511540912951799364, −9.252688494007028027432748092040, −8.532566276436728188030953233515, −7.38586817824468723028981525903, −6.74945552899266350144615899582, −4.79635229167059284675856483530, −3.94626396319876935588304232935, −2.71079382891300965046633340424, 1.68399124395867391503759872758, 3.40119169248587176864998601308, 4.24154280934798370694093078539, 5.77035512122828915916735729557, 7.29814015071637522751620388243, 8.326389774921783239589592292707, 9.012348326274563911151486306308, 10.38104178391186230083372223714, 11.35787851763550067233293265689, 12.51833497632081383199654237128

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