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Properties

Label 2-210-35.4-c1-0-1
Degree $2$
Conductor $210$
Sign $0.386 - 0.922i$
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  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.806 + 2.08i)5-s + 0.999·6-s + (2.24 − 1.40i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.344 − 2.20i)10-s + (−0.838 + 1.45i)11-s + (−0.866 + 0.499i)12-s + 4.48i·13-s + (−1.24 + 2.33i)14-s + (1.74 − 1.40i)15-s + (−0.5 − 0.866i)16-s + (6.59 + 3.80i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.360 + 0.932i)5-s + 0.408·6-s + (0.847 − 0.530i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.108 − 0.698i)10-s + (−0.252 + 0.437i)11-s + (−0.249 + 0.144i)12-s + 1.24i·13-s + (−0.331 + 0.624i)14-s + (0.449 − 0.362i)15-s + (−0.125 − 0.216i)16-s + (1.59 + 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641172 + 0.426344i\)
\(L(\frac12)\) \(\approx\) \(0.641172 + 0.426344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.806 - 2.08i)T \)
7 \( 1 + (-2.24 + 1.40i)T \)
good11 \( 1 + (0.838 - 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.48iT - 13T^{2} \)
17 \( 1 + (-6.59 - 3.80i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.24 - 3.88i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.417 - 0.241i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 + (-1.74 + 3.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.417 + 0.241i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (-9.91 + 5.72i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.29 + 4.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.88 + 4.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.28 - 9.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.90 + 1.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.25 - 3.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.87iT - 83T^{2} \)
89 \( 1 + (1.67 + 2.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.7iT - 97T^{2} \)
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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

−12.14112444173894958952568314683, −11.53707852398144743713744056392, −10.50722585811306180003939996086, −9.894914191365453824936108492730, −8.215465027583351862710395257938, −7.51728661024886598147560171482, −6.66407796665781897246134099877, −5.44499587561760685304245731847, −3.89503626144813720107609365513, −1.73091716192057246190613261842, 0.956375160310457147250412355678, 3.14408311242925081863406971467, 4.90131417149151003113657237253, 5.60009753283197567992858184068, 7.52038960636109488974260072215, 8.270339223359885937676684307530, 9.246944720828834738486767163065, 10.26097812357279607142016135780, 11.31789674486595843691808198842, 11.98311015658376537645759665509

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