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Properties

Label 2-210-15.8-c1-0-7
Degree $2$
Conductor $210$
Sign $0.837 - 0.546i$
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 + (1.33 − 1.10i)3-s + 1.00i·4-s + (0.953 + 2.02i)5-s + (1.72 + 0.162i)6-s + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + (0.559 − 2.94i)9-s + (−0.755 + 2.10i)10-s + 0.780i·11-s + (1.10 + 1.33i)12-s + (−3.85 − 3.85i)13-s − 1.00·14-s + (3.50 + 1.64i)15-s − 1.00·16-s + (2.97 + 2.97i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.770 − 0.637i)3-s + 0.500i·4-s + (0.426 + 0.904i)5-s + (0.703 + 0.0662i)6-s + (−0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.186 − 0.982i)9-s + (−0.238 + 0.665i)10-s + 0.235i·11-s + (0.318 + 0.385i)12-s + (−1.06 − 1.06i)13-s − 0.267·14-s + (0.905 + 0.424i)15-s − 0.250·16-s + (0.721 + 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.837 - 0.546i$
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: \((2,\ 210,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83064 + 0.545030i\)
\(L(\frac12)\) \(\approx\) \(1.83064 + 0.545030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 + 1.10i)T \)
5 \( 1 + (-0.953 - 2.02i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 0.780iT - 11T^{2} \)
13 \( 1 + (3.85 + 3.85i)T + 13iT^{2} \)
17 \( 1 + (-2.97 - 2.97i)T + 17iT^{2} \)
19 \( 1 + 5.79iT - 19T^{2} \)
23 \( 1 + (-1.74 + 1.74i)T - 23iT^{2} \)
29 \( 1 + 3.33T + 29T^{2} \)
31 \( 1 - 1.20T + 31T^{2} \)
37 \( 1 + (6.28 - 6.28i)T - 37iT^{2} \)
41 \( 1 + 0.410iT - 41T^{2} \)
43 \( 1 + (0.397 + 0.397i)T + 43iT^{2} \)
47 \( 1 + (4.50 + 4.50i)T + 47iT^{2} \)
53 \( 1 + (7.69 - 7.69i)T - 53iT^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 9.11T + 61T^{2} \)
67 \( 1 + (-4.95 + 4.95i)T - 67iT^{2} \)
71 \( 1 - 1.88iT - 71T^{2} \)
73 \( 1 + (8.48 + 8.48i)T + 73iT^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (3.60 - 3.60i)T - 83iT^{2} \)
89 \( 1 - 18.5T + 89T^{2} \)
97 \( 1 + (8.82 - 8.82i)T - 97iT^{2} \)
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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

−12.79442158948436264203832096749, −11.82131834901739334568131065543, −10.38875079641847290277386533196, −9.434331464158222691572197666578, −8.198066936951507648992284897241, −7.24336195786975987659325105590, −6.50473187837178560857014409351, −5.25363507466063378890496293801, −3.39300971222922997115738619649, −2.45848742429476388069687027578, 1.94950869492994276523726408722, 3.51231934183477611069049825391, 4.63463895328328846728541334895, 5.58128085635438878021751230058, 7.29542354192581961119968297492, 8.587498192663668750623463623775, 9.656252497975660190597644028106, 9.973014923058095216597386173908, 11.41462281526852630497260702444, 12.41083547151104607246470688876

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