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Properties

Label 2-210-105.62-c2-0-18
Degree $2$
Conductor $210$
Sign $0.442 + 0.896i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (2.09 − 2.14i)3-s − 2i·4-s + (1.13 + 4.86i)5-s + (−0.0523 − 4.24i)6-s + (6.60 + 2.31i)7-s + (−2 − 2i)8-s + (−0.222 − 8.99i)9-s + (6.00 + 3.73i)10-s − 16.9i·11-s + (−4.29 − 4.18i)12-s + (10.2 + 10.2i)13-s + (8.91 − 4.29i)14-s + (12.8 + 7.76i)15-s − 4·16-s + (8.79 + 8.79i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.698 − 0.715i)3-s − 0.5i·4-s + (0.227 + 0.973i)5-s + (−0.00872 − 0.707i)6-s + (0.943 + 0.330i)7-s + (−0.250 − 0.250i)8-s + (−0.0246 − 0.999i)9-s + (0.600 + 0.373i)10-s − 1.54i·11-s + (−0.357 − 0.349i)12-s + (0.786 + 0.786i)13-s + (0.637 − 0.306i)14-s + (0.855 + 0.517i)15-s − 0.250·16-s + (0.517 + 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.442 + 0.896i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.442 + 0.896i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.28811 - 1.42229i\)
\(L(\frac12)\) \(\approx\) \(2.28811 - 1.42229i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (-2.09 + 2.14i)T \)
5 \( 1 + (-1.13 - 4.86i)T \)
7 \( 1 + (-6.60 - 2.31i)T \)
good11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 + (-10.2 - 10.2i)T + 169iT^{2} \)
17 \( 1 + (-8.79 - 8.79i)T + 289iT^{2} \)
19 \( 1 + 24.7T + 361T^{2} \)
23 \( 1 + (19.2 + 19.2i)T + 529iT^{2} \)
29 \( 1 + 1.67T + 841T^{2} \)
31 \( 1 - 36.8iT - 961T^{2} \)
37 \( 1 + (-40.5 - 40.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 0.885T + 1.68e3T^{2} \)
43 \( 1 + (9.87 - 9.87i)T - 1.84e3iT^{2} \)
47 \( 1 + (33.7 + 33.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-11.9 - 11.9i)T + 2.80e3iT^{2} \)
59 \( 1 + 50.5iT - 3.48e3T^{2} \)
61 \( 1 - 80.6iT - 3.72e3T^{2} \)
67 \( 1 + (4.46 + 4.46i)T + 4.48e3iT^{2} \)
71 \( 1 - 137. iT - 5.04e3T^{2} \)
73 \( 1 + (53.3 + 53.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 127. iT - 6.24e3T^{2} \)
83 \( 1 + (60.0 - 60.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 51.3iT - 7.92e3T^{2} \)
97 \( 1 + (-0.274 + 0.274i)T - 9.40e3iT^{2} \)
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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

−11.88023574372995629541578611518, −11.19276390689977496699564475154, −10.29462861825411091427935937192, −8.751878020290954402904118266946, −8.161864438706547808655126389130, −6.58864030601463731144456659574, −5.92876070471355541217909175452, −4.00785835594296512708908085941, −2.84333537798040057085883641480, −1.60782938597115253951264239897, 2.01722566978591461249623104488, 4.00757406963784240687284185804, 4.69742435216685296060002840677, 5.73480520702214876327085825291, 7.60579856141270659235178322068, 8.149773796062235416814510379837, 9.268711962028352664108087946838, 10.17946825905848260899723977894, 11.40060084527115596943548247225, 12.63848318760768739509106130378

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