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Properties

Label 2-210-105.32-c1-0-14
Degree $2$
Conductor $210$
Sign $0.518 + 0.854i$
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 + (−0.125 − 1.72i)3-s + (0.866 + 0.499i)4-s + (−1.36 − 1.77i)5-s + (0.325 − 1.70i)6-s + (2.62 − 0.301i)7-s + (0.707 + 0.707i)8-s + (−2.96 + 0.433i)9-s + (−0.854 − 2.06i)10-s + (−1.49 − 0.860i)11-s + (0.755 − 1.55i)12-s + (1.71 − 1.71i)13-s + (2.61 + 0.389i)14-s + (−2.89 + 2.57i)15-s + (0.500 + 0.866i)16-s + (0.641 + 2.39i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.0724 − 0.997i)3-s + (0.433 + 0.249i)4-s + (−0.608 − 0.793i)5-s + (0.133 − 0.694i)6-s + (0.993 − 0.113i)7-s + (0.249 + 0.249i)8-s + (−0.989 + 0.144i)9-s + (−0.270 − 0.653i)10-s + (−0.449 − 0.259i)11-s + (0.217 − 0.449i)12-s + (0.474 − 0.474i)13-s + (0.699 + 0.104i)14-s + (−0.747 + 0.664i)15-s + (0.125 + 0.216i)16-s + (0.155 + 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.518 + 0.854i$
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: \((2,\ 210,\ (\ :1/2),\ 0.518 + 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43521 - 0.807981i\)
\(L(\frac12)\) \(\approx\) \(1.43521 - 0.807981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.125 + 1.72i)T \)
5 \( 1 + (1.36 + 1.77i)T \)
7 \( 1 + (-2.62 + 0.301i)T \)
good11 \( 1 + (1.49 + 0.860i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.71 + 1.71i)T - 13iT^{2} \)
17 \( 1 + (-0.641 - 2.39i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.74 + 2.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.97 - 7.36i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + (2.74 - 4.75i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.29 - 4.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.76iT - 41T^{2} \)
43 \( 1 + (1.33 - 1.33i)T - 43iT^{2} \)
47 \( 1 + (5.84 + 1.56i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-13.1 + 3.51i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.45 - 9.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.22 + 7.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.2 - 3.54i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (0.849 + 3.17i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.97 + 1.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.66 + 1.66i)T + 83iT^{2} \)
89 \( 1 + (1.63 + 2.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.1 - 11.1i)T + 97iT^{2} \)
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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

−12.15998389401015038228320498528, −11.68266895089616933634012143307, −10.69982815765222589096284419534, −8.770731868309333676452526365375, −7.970030913823788517626913851375, −7.29226655866780954432649157271, −5.74862717951624701158415132001, −4.97219317292033641644406832565, −3.39288137272412819646429160347, −1.44537296441089989104413979561, 2.64079270328394043939350602852, 3.96140404889282519230610740676, 4.85146566249993680708167587888, 6.05743321469169754874733603549, 7.46357353375813939655322346380, 8.528853280854775284123973377932, 9.954935031415249524765213745981, 10.79051804519857120057063624172, 11.49594203489087125542716423374, 12.18691982663066711358414952497

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