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Properties

Label 2-210-21.17-c1-0-10
Degree $2$
Conductor $210$
Sign $0.412 + 0.910i$
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  + (0.866 − 0.5i)2-s + (1.20 − 1.24i)3-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.421 − 1.68i)6-s + (0.0551 + 2.64i)7-s − 0.999i·8-s + (−0.0969 − 2.99i)9-s + (−0.866 − 0.499i)10-s + (0.167 + 0.0969i)11-s + (−0.475 − 1.66i)12-s + 1.54i·13-s + (1.37 + 2.26i)14-s + (−1.68 − 0.421i)15-s + (−0.5 − 0.866i)16-s + (−0.264 + 0.458i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.695 − 0.718i)3-s + (0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.171 − 0.685i)6-s + (0.0208 + 0.999i)7-s − 0.353i·8-s + (−0.0323 − 0.999i)9-s + (−0.273 − 0.158i)10-s + (0.0506 + 0.0292i)11-s + (−0.137 − 0.480i)12-s + 0.429i·13-s + (0.366 + 0.604i)14-s + (−0.433 − 0.108i)15-s + (−0.125 − 0.216i)16-s + (−0.0642 + 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.412 + 0.910i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.412 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61210 - 1.03927i\)
\(L(\frac12)\) \(\approx\) \(1.61210 - 1.03927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-1.20 + 1.24i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.0551 - 2.64i)T \)
good11 \( 1 + (-0.167 - 0.0969i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.54iT - 13T^{2} \)
17 \( 1 + (0.264 - 0.458i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.53 - 3.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.87iT - 29T^{2} \)
31 \( 1 + (-8.02 - 4.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.881 + 1.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.91T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (4.90 + 8.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0562 + 0.0324i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.01 - 10.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.71 + 2.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.29iT - 71T^{2} \)
73 \( 1 + (7.00 + 4.04i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.38 + 5.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.11T + 83T^{2} \)
89 \( 1 + (8.18 + 14.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.24iT - 97T^{2} \)
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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

−12.36786265388677070767709867381, −11.67162015127982795295033794590, −10.31893063142655810403022684713, −8.949985493679545099542379718424, −8.429771200331550154933456490599, −6.96601648387910926907934070695, −5.96838456560774949158931460850, −4.55149694617860062668182511826, −3.10933225622110556791470746111, −1.80574913695541116653285857219, 2.74916520334739229163935109294, 3.95788377439320553664153894261, 4.80860895756702602259898992439, 6.42166665387492749315419040791, 7.53719844544378211786752276794, 8.375862601039038475034602021965, 9.686022152607282132054331725054, 10.66855695527352706029489848333, 11.40065408342941960733764503464, 12.94128003485046642292235928830

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