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Properties

Label 2-210-35.33-c1-0-3
Degree $2$
Conductor $210$
Sign $-0.0732 - 0.997i$
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  + (0.258 + 0.965i)2-s + (0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + (1.04 + 1.97i)5-s + i·6-s + (−2.55 + 0.703i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−1.64 + 1.51i)10-s + (0.989 + 1.71i)11-s + (−0.965 + 0.258i)12-s + (2.19 − 2.19i)13-s + (−1.33 − 2.28i)14-s + (0.493 + 2.18i)15-s + (0.500 − 0.866i)16-s + (1.19 − 4.44i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.465 + 0.884i)5-s + 0.408i·6-s + (−0.963 + 0.265i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.519 + 0.480i)10-s + (0.298 + 0.516i)11-s + (−0.278 + 0.0747i)12-s + (0.608 − 0.608i)13-s + (−0.358 − 0.609i)14-s + (0.127 + 0.563i)15-s + (0.125 − 0.216i)16-s + (0.288 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0732 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0732 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.0732 - 0.997i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.0732 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01176 + 1.08877i\)
\(L(\frac12)\) \(\approx\) \(1.01176 + 1.08877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-1.04 - 1.97i)T \)
7 \( 1 + (2.55 - 0.703i)T \)
good11 \( 1 + (-0.989 - 1.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.19 + 2.19i)T - 13iT^{2} \)
17 \( 1 + (-1.19 + 4.44i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.10 - 3.65i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.68 + 1.52i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.94iT - 29T^{2} \)
31 \( 1 + (1.50 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.717 + 2.67i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 + (6.33 + 6.33i)T + 43iT^{2} \)
47 \( 1 + (-5.87 + 1.57i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.95 + 11.0i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.10 - 3.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.63 - 5.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.32 + 1.42i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.86T + 71T^{2} \)
73 \( 1 + (14.7 + 3.93i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.21 + 1.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.52 - 9.52i)T - 83iT^{2} \)
89 \( 1 + (3.09 - 5.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.48 - 1.48i)T + 97iT^{2} \)
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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

−13.02965175827758731169756503793, −11.77648063279563229870822871601, −10.32492820246270348070572730181, −9.671358058527167496964791730021, −8.660368104049676015977939056985, −7.38175234578209006383732393230, −6.55390845272620140033135445226, −5.51949027895919201526358520851, −3.81246592987408748229006147815, −2.69422637750338143520831217646, 1.39044946075330768687028721148, 3.13962501745289712786095761050, 4.28327229320292651494618707525, 5.76096581231169082755431829937, 6.91636501357514345746927197722, 8.696090847056041889285923699115, 9.013823162990260804832332139640, 10.12904326914653731818755477745, 11.12901923380013726197446022362, 12.43150534331203388530775124285

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