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Properties

Label 2-210-15.2-c1-0-11
Degree $2$
Conductor $210$
Sign $-0.882 + 0.470i$
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 + (−0.799 − 1.53i)3-s − 1.00i·4-s + (−1.91 − 1.15i)5-s + (−1.65 − 0.521i)6-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (−1.72 + 2.45i)9-s + (−2.17 + 0.536i)10-s + 1.70i·11-s + (−1.53 + 0.799i)12-s + (0.921 − 0.921i)13-s − 1.00·14-s + (−0.245 + 3.86i)15-s − 1.00·16-s + (4.76 − 4.76i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.461 − 0.887i)3-s − 0.500i·4-s + (−0.856 − 0.516i)5-s + (−0.674 − 0.212i)6-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.574 + 0.818i)9-s + (−0.686 + 0.169i)10-s + 0.514i·11-s + (−0.443 + 0.230i)12-s + (0.255 − 0.255i)13-s − 0.267·14-s + (−0.0633 + 0.997i)15-s − 0.250·16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.882 + 0.470i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.882 + 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247686 - 0.990323i\)
\(L(\frac12)\) \(\approx\) \(0.247686 - 0.990323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.799 + 1.53i)T \)
5 \( 1 + (1.91 + 1.15i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - 1.70iT - 11T^{2} \)
13 \( 1 + (-0.921 + 0.921i)T - 13iT^{2} \)
17 \( 1 + (-4.76 + 4.76i)T - 17iT^{2} \)
19 \( 1 + 5.94iT - 19T^{2} \)
23 \( 1 + (2.49 + 2.49i)T + 23iT^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 3.40T + 31T^{2} \)
37 \( 1 + (1.02 + 1.02i)T + 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (-8.17 + 8.17i)T - 43iT^{2} \)
47 \( 1 + (-0.436 + 0.436i)T - 47iT^{2} \)
53 \( 1 + (-6.87 - 6.87i)T + 53iT^{2} \)
59 \( 1 + 0.686T + 59T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 + (-1.03 - 1.03i)T + 67iT^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 + (4.59 - 4.59i)T - 73iT^{2} \)
79 \( 1 + 7.19iT - 79T^{2} \)
83 \( 1 + (6.64 + 6.64i)T + 83iT^{2} \)
89 \( 1 - 2.25T + 89T^{2} \)
97 \( 1 + (-13.2 - 13.2i)T + 97iT^{2} \)
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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

−12.05000674049063289897863060240, −11.37705386916233068148437501951, −10.31157798897687844926451269407, −8.970346294899924368520301587966, −7.69904702974162708158532306095, −6.88763649750061197528188261432, −5.48096203611644555714227191197, −4.45284564044072430671537384923, −2.84028342639252256624940587162, −0.827305020372959768805519071667, 3.38102466287084538668901232531, 4.03764387380714781838129291071, 5.58561973362547419135111400396, 6.31471737127748195671023291577, 7.74747795650766654346786571290, 8.669032242151512933385668325116, 10.04851997268778115678081795310, 10.88043113622103659462577913632, 11.94442257831837143052740460274, 12.47468588658565075298906690994

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