Properties

Label 2-210-105.23-c1-0-11
Degree $2$
Conductor $210$
Sign $0.701 + 0.712i$
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 + (−0.0744 − 1.73i)3-s + (0.866 − 0.499i)4-s + (1.68 + 1.47i)5-s + (−0.519 − 1.65i)6-s + (2.42 + 1.05i)7-s + (0.707 − 0.707i)8-s + (−2.98 + 0.257i)9-s + (2.00 + 0.989i)10-s + (−2.50 + 1.44i)11-s + (−0.929 − 1.46i)12-s + (−4.54 − 4.54i)13-s + (2.61 + 0.393i)14-s + (2.42 − 3.01i)15-s + (0.500 − 0.866i)16-s + (0.372 − 1.39i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.0429 − 0.999i)3-s + (0.433 − 0.249i)4-s + (0.751 + 0.659i)5-s + (−0.212 − 0.674i)6-s + (0.916 + 0.399i)7-s + (0.249 − 0.249i)8-s + (−0.996 + 0.0858i)9-s + (0.634 + 0.313i)10-s + (−0.754 + 0.435i)11-s + (−0.268 − 0.421i)12-s + (−1.26 − 1.26i)13-s + (0.699 + 0.105i)14-s + (0.626 − 0.779i)15-s + (0.125 − 0.216i)16-s + (0.0904 − 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.701 + 0.712i$
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: \((2,\ 210,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73035 - 0.724954i\)
\(L(\frac12)\) \(\approx\) \(1.73035 - 0.724954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0744 + 1.73i)T \)
5 \( 1 + (-1.68 - 1.47i)T \)
7 \( 1 + (-2.42 - 1.05i)T \)
good11 \( 1 + (2.50 - 1.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.54 + 4.54i)T + 13iT^{2} \)
17 \( 1 + (-0.372 + 1.39i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.42 - 0.820i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.27 - 4.76i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
31 \( 1 + (-0.155 - 0.269i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.756 - 2.82i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.54iT - 41T^{2} \)
43 \( 1 + (-6.09 - 6.09i)T + 43iT^{2} \)
47 \( 1 + (-5.65 + 1.51i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.518 + 0.138i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.20 + 5.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.348 + 0.603i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.807 - 0.216i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.72iT - 71T^{2} \)
73 \( 1 + (0.0483 - 0.180i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.11 + 4.68i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.26 + 2.26i)T - 83iT^{2} \)
89 \( 1 + (3.60 - 6.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.75 + 4.75i)T - 97iT^{2} \)
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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

−12.43743778427099496022824030642, −11.44038731217673168666057202243, −10.59425196056669468477826259200, −9.433821136823509740620767725864, −7.71998960296676014868595036477, −7.29908613101691874726425120184, −5.68420415663166061851137247397, −5.25178205155545780339668794747, −2.93171190484942800659232472414, −1.96789334743234967911382330016, 2.34621396470250715324734986377, 4.21165339734964043864699220925, 4.96338582959582872125956536008, 5.82355459305078981055200090775, 7.40476131856536143330010731916, 8.645005852741994029596121425231, 9.590768328662679540765954511621, 10.65017844801048947113310960705, 11.47987507036034024978811197807, 12.54496163440312769389954996373

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