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Properties

Label 2-210-105.32-c1-0-13
Degree $2$
Conductor $210$
Sign $-0.529 + 0.848i$
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  + (−0.965 − 0.258i)2-s + (1.03 − 1.38i)3-s + (0.866 + 0.499i)4-s + (−2.22 + 0.221i)5-s + (−1.36 + 1.06i)6-s + (−0.736 − 2.54i)7-s + (−0.707 − 0.707i)8-s + (−0.839 − 2.88i)9-s + (2.20 + 0.361i)10-s + (−1.83 − 1.05i)11-s + (1.59 − 0.680i)12-s + (3.28 − 3.28i)13-s + (0.0535 + 2.64i)14-s + (−2.00 + 3.31i)15-s + (0.500 + 0.866i)16-s + (0.375 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.600 − 0.799i)3-s + (0.433 + 0.249i)4-s + (−0.995 + 0.0991i)5-s + (−0.556 + 0.436i)6-s + (−0.278 − 0.960i)7-s + (−0.249 − 0.249i)8-s + (−0.279 − 0.960i)9-s + (0.697 + 0.114i)10-s + (−0.553 − 0.319i)11-s + (0.459 − 0.196i)12-s + (0.910 − 0.910i)13-s + (0.0143 + 0.706i)14-s + (−0.517 + 0.855i)15-s + (0.125 + 0.216i)16-s + (0.0910 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.529 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378738 - 0.682485i\)
\(L(\frac12)\) \(\approx\) \(0.378738 - 0.682485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.03 + 1.38i)T \)
5 \( 1 + (2.22 - 0.221i)T \)
7 \( 1 + (0.736 + 2.54i)T \)
good11 \( 1 + (1.83 + 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.28 + 3.28i)T - 13iT^{2} \)
17 \( 1 + (-0.375 - 1.40i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.72 - 2.72i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.47 + 5.50i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.20T + 29T^{2} \)
31 \( 1 + (0.528 - 0.914i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.66 - 6.22i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 1.86iT - 41T^{2} \)
43 \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \)
47 \( 1 + (-10.7 - 2.87i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.47 + 1.73i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.63 - 4.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.11 - 7.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.10 + 2.43i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-0.323 - 1.20i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-13.8 + 7.99i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.47 + 9.47i)T + 83iT^{2} \)
89 \( 1 + (-0.199 - 0.346i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.63 + 8.63i)T + 97iT^{2} \)
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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

−12.16662030764880380129020510525, −10.81854362307011017835469614766, −10.34328381988373472183451527080, −8.571059914494251060663998513582, −8.227432138849036525900648801370, −7.22412842192559053044201373121, −6.27785496277844690877276184489, −3.99990747272664498642701232991, −2.91004001335209157102247716658, −0.791329182639766842467388363543, 2.51882619374347087753587712552, 3.95389631460493766708881468455, 5.27204130054753272647600399096, 6.81192335164126539845139068410, 8.051960680585540522677328472826, 8.790208963878174665646753183373, 9.461482805623573197089569674848, 10.73859098924276832943398241318, 11.43998220476826582544187980001, 12.53840458280687664411751449987

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