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Properties

Label 2-210-105.23-c1-0-15
Degree $2$
Conductor $210$
Sign $0.634 + 0.772i$
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 + (1.46 − 0.924i)3-s + (0.866 − 0.499i)4-s + (−0.480 − 2.18i)5-s + (1.17 − 1.27i)6-s + (−0.998 + 2.45i)7-s + (0.707 − 0.707i)8-s + (1.29 − 2.70i)9-s + (−1.02 − 1.98i)10-s + (−5.16 + 2.98i)11-s + (0.806 − 1.53i)12-s + (2.36 + 2.36i)13-s + (−0.330 + 2.62i)14-s + (−2.72 − 2.75i)15-s + (0.500 − 0.866i)16-s + (0.245 − 0.914i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.845 − 0.533i)3-s + (0.433 − 0.249i)4-s + (−0.214 − 0.976i)5-s + (0.480 − 0.519i)6-s + (−0.377 + 0.926i)7-s + (0.249 − 0.249i)8-s + (0.430 − 0.902i)9-s + (−0.325 − 0.627i)10-s + (−1.55 + 0.899i)11-s + (0.232 − 0.442i)12-s + (0.656 + 0.656i)13-s + (−0.0883 + 0.701i)14-s + (−0.702 − 0.711i)15-s + (0.125 − 0.216i)16-s + (0.0594 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83609 - 0.868175i\)
\(L(\frac12)\) \(\approx\) \(1.83609 - 0.868175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 + 0.924i)T \)
5 \( 1 + (0.480 + 2.18i)T \)
7 \( 1 + (0.998 - 2.45i)T \)
good11 \( 1 + (5.16 - 2.98i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.36 - 2.36i)T + 13iT^{2} \)
17 \( 1 + (-0.245 + 0.914i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.22 - 2.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0558 + 0.208i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.38T + 29T^{2} \)
31 \( 1 + (-1.28 - 2.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.352 + 1.31i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.15iT - 41T^{2} \)
43 \( 1 + (6.02 + 6.02i)T + 43iT^{2} \)
47 \( 1 + (3.61 - 0.969i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.98 + 1.60i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.02 - 3.50i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.70 - 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.28 + 2.48i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (-3.43 + 12.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.874 - 0.504i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.84 + 7.84i)T - 83iT^{2} \)
89 \( 1 + (-0.188 + 0.327i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.42 + 3.42i)T - 97iT^{2} \)
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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

−12.35487598264072782155722866806, −11.85455398909634597729316716477, −10.15849328485372112464277910704, −9.178814469852685812934364743950, −8.234990250190960893780449578908, −7.23289901706085637501595824986, −5.80912479054388226468941458704, −4.71398214349621930376346006521, −3.22542845136698720499352661524, −1.90002421840409933777892832710, 2.93581876782604510549634448609, 3.45628009249572902854714201236, 4.93138077491955313823461757995, 6.31665784438073919417944238379, 7.62000304960492827396900835920, 8.132806015383410671277184730464, 9.846221390559177906811998020733, 10.64195336881679338184165897636, 11.24458451817290483232383696247, 13.02973416586633804579216910520

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