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Properties

Label 2-210-105.53-c1-0-5
Degree $2$
Conductor $210$
Sign $0.986 - 0.162i$
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  + (−0.258 + 0.965i)2-s + (−1.69 + 0.378i)3-s + (−0.866 − 0.499i)4-s + (0.593 − 2.15i)5-s + (0.0717 − 1.73i)6-s + (2.64 − 0.113i)7-s + (0.707 − 0.707i)8-s + (2.71 − 1.27i)9-s + (1.92 + 1.13i)10-s + (−3.54 − 2.04i)11-s + (1.65 + 0.517i)12-s + (3.69 + 3.69i)13-s + (−0.574 + 2.58i)14-s + (−0.186 + 3.86i)15-s + (0.500 + 0.866i)16-s + (7.20 − 1.93i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.975 + 0.218i)3-s + (−0.433 − 0.249i)4-s + (0.265 − 0.964i)5-s + (0.0292 − 0.706i)6-s + (0.999 − 0.0427i)7-s + (0.249 − 0.249i)8-s + (0.904 − 0.426i)9-s + (0.609 + 0.357i)10-s + (−1.06 − 0.616i)11-s + (0.477 + 0.149i)12-s + (1.02 + 1.02i)13-s + (−0.153 + 0.690i)14-s + (−0.0482 + 0.998i)15-s + (0.125 + 0.216i)16-s + (1.74 − 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.986 - 0.162i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.986 - 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.907592 + 0.0741129i\)
\(L(\frac12)\) \(\approx\) \(0.907592 + 0.0741129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (1.69 - 0.378i)T \)
5 \( 1 + (-0.593 + 2.15i)T \)
7 \( 1 + (-2.64 + 0.113i)T \)
good11 \( 1 + (3.54 + 2.04i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.69 - 3.69i)T + 13iT^{2} \)
17 \( 1 + (-7.20 + 1.93i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.12 + 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.06 + 0.284i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 1.48T + 29T^{2} \)
31 \( 1 + (-2.62 + 4.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.34 + 1.43i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.74iT - 41T^{2} \)
43 \( 1 + (-1.27 - 1.27i)T + 43iT^{2} \)
47 \( 1 + (0.796 - 2.97i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.47 - 5.50i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.09 - 8.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.814 + 1.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.909 - 3.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 5.89iT - 71T^{2} \)
73 \( 1 + (5.38 - 1.44i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.18 - 4.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.31 - 6.31i)T - 83iT^{2} \)
89 \( 1 + (-6.71 - 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.21 - 5.21i)T - 97iT^{2} \)
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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

−12.28113922119191019806765121195, −11.44602808282655534388822364296, −10.44033207520884433394263395924, −9.375579428220257811404098612982, −8.333759760275878039370501664656, −7.38583323852326003409987259180, −5.80432246414123165332357124577, −5.33363541393493799891991061284, −4.20171551550435632133974117913, −1.15487606928045393639366501820, 1.58237961927791764783206382900, 3.32140861130207095764731067434, 5.06309270433078581154587581712, 5.89778705330190936052500655602, 7.47067064706113196704840965066, 8.122013008190373107364458507788, 10.12509306471288350757460793104, 10.38217528432872353629859187599, 11.28045049779544932640460937385, 12.14218591310631467840000364280

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