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Properties

Label 2-210-105.83-c2-0-10
Degree $2$
Conductor $210$
Sign $-0.587 - 0.809i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (−2.98 + 0.282i)3-s + 2i·4-s + (3.28 − 3.77i)5-s + (−3.26 − 2.70i)6-s + (−3.67 + 5.95i)7-s + (−2 + 2i)8-s + (8.84 − 1.68i)9-s + (7.05 − 0.487i)10-s + 19.5i·11-s + (−0.565 − 5.97i)12-s + (−2.90 + 2.90i)13-s + (−9.63 + 2.28i)14-s + (−8.74 + 12.1i)15-s − 4·16-s + (−16.3 + 16.3i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.995 + 0.0942i)3-s + 0.5i·4-s + (0.656 − 0.754i)5-s + (−0.544 − 0.450i)6-s + (−0.525 + 0.850i)7-s + (−0.250 + 0.250i)8-s + (0.982 − 0.187i)9-s + (0.705 − 0.0487i)10-s + 1.77i·11-s + (−0.0471 − 0.497i)12-s + (−0.223 + 0.223i)13-s + (−0.688 + 0.162i)14-s + (−0.582 + 0.812i)15-s − 0.250·16-s + (−0.959 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.587 - 0.809i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.587 - 0.809i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.558538 + 1.09526i\)
\(L(\frac12)\) \(\approx\) \(0.558538 + 1.09526i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (2.98 - 0.282i)T \)
5 \( 1 + (-3.28 + 3.77i)T \)
7 \( 1 + (3.67 - 5.95i)T \)
good11 \( 1 - 19.5iT - 121T^{2} \)
13 \( 1 + (2.90 - 2.90i)T - 169iT^{2} \)
17 \( 1 + (16.3 - 16.3i)T - 289iT^{2} \)
19 \( 1 + 8.66T + 361T^{2} \)
23 \( 1 + (-6.73 + 6.73i)T - 529iT^{2} \)
29 \( 1 - 31.3T + 841T^{2} \)
31 \( 1 - 39.4iT - 961T^{2} \)
37 \( 1 + (25.1 - 25.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 58.9T + 1.68e3T^{2} \)
43 \( 1 + (-10.5 - 10.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-29.2 + 29.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-10.3 + 10.3i)T - 2.80e3iT^{2} \)
59 \( 1 + 42.5iT - 3.48e3T^{2} \)
61 \( 1 + 45.1iT - 3.72e3T^{2} \)
67 \( 1 + (-89.3 + 89.3i)T - 4.48e3iT^{2} \)
71 \( 1 - 47.3iT - 5.04e3T^{2} \)
73 \( 1 + (89.3 - 89.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 41.4iT - 6.24e3T^{2} \)
83 \( 1 + (44.9 + 44.9i)T + 6.88e3iT^{2} \)
89 \( 1 - 4.80iT - 7.92e3T^{2} \)
97 \( 1 + (2.01 + 2.01i)T + 9.40e3iT^{2} \)
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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

−12.64167417598791093464550142665, −11.95510093919304171409684833352, −10.47257598592159628069160702901, −9.569074006010151681703177918241, −8.575808512321656403153549046925, −6.91564077337082676651490101086, −6.24664697345056652268412687258, −5.09495523104142252963896438216, −4.39694153175102520068161980177, −2.05173488731542428434809544692, 0.65570629848420308370985754999, 2.75400150311778457219234467426, 4.15270147461297126304212263111, 5.62733791623498617165528775997, 6.35354964297605208194838560334, 7.31440301042805742727148716689, 9.205165670275871895939605253489, 10.31466342106207210585303456744, 10.91832590749243142747828890863, 11.54291805714622826255573933484

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