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Properties

Label 2-210-35.27-c1-0-0
Degree $2$
Conductor $210$
Sign $0.682 - 0.731i$
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 + (0.707 + 0.707i)3-s + 1.00i·4-s + (1.52 + 1.63i)5-s − 1.00i·6-s + (−1.23 + 2.33i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (0.0743 − 2.23i)10-s − 5.73·11-s + (−0.707 + 0.707i)12-s + (3.41 + 3.41i)13-s + (2.52 − 0.781i)14-s + (−0.0743 + 2.23i)15-s − 1.00·16-s + (2.57 − 2.57i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.683 + 0.730i)5-s − 0.408i·6-s + (−0.466 + 0.884i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.0234 − 0.706i)10-s − 1.72·11-s + (−0.204 + 0.204i)12-s + (0.946 + 0.946i)13-s + (0.675 − 0.208i)14-s + (−0.0191 + 0.577i)15-s − 0.250·16-s + (0.624 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991195 + 0.430896i\)
\(L(\frac12)\) \(\approx\) \(0.991195 + 0.430896i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.52 - 1.63i)T \)
7 \( 1 + (1.23 - 2.33i)T \)
good11 \( 1 + 5.73T + 11T^{2} \)
13 \( 1 + (-3.41 - 3.41i)T + 13iT^{2} \)
17 \( 1 + (-2.57 + 2.57i)T - 17iT^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 + (-6.46 + 6.46i)T - 23iT^{2} \)
29 \( 1 + 3.47iT - 29T^{2} \)
31 \( 1 - 0.469iT - 31T^{2} \)
37 \( 1 + (-0.574 - 0.574i)T + 37iT^{2} \)
41 \( 1 + 1.03iT - 41T^{2} \)
43 \( 1 + (-6.17 + 6.17i)T - 43iT^{2} \)
47 \( 1 + (3.85 - 3.85i)T - 47iT^{2} \)
53 \( 1 + (1.85 - 1.85i)T - 53iT^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 8.53iT - 61T^{2} \)
67 \( 1 + (4.46 + 4.46i)T + 67iT^{2} \)
71 \( 1 - 5.13T + 71T^{2} \)
73 \( 1 + (-7.80 - 7.80i)T + 73iT^{2} \)
79 \( 1 + 4.16iT - 79T^{2} \)
83 \( 1 + (-1.77 - 1.77i)T + 83iT^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + (0.0119 - 0.0119i)T - 97iT^{2} \)
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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

−12.50621338368658458612977822477, −11.19298826334173827789515113867, −10.50752319026170954277385999332, −9.568588503819607339036205187117, −8.854041579286613083827001693905, −7.66998138292488275150389284549, −6.35586527373938185953308173132, −5.08065086276964956405367069337, −3.17507236234488831582540962300, −2.38028946665391801598917820875, 1.15119552158456936154031054581, 3.17818393567404962099542291239, 5.14053288945735240046990325773, 6.04609041069633131986005801902, 7.46986901040519428710034079259, 8.089217853506441571914779966670, 9.207120281178590333496172074039, 10.16244979172090137179127964423, 10.89375251655355800350104575220, 12.76713621572094100520127967865

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