Properties

Degree $2$
Conductor $210$
Sign $-0.441 - 0.897i$
Motivic weight $1$
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 + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.86 + 1.23i)5-s − 0.999·6-s + (−1.73 + 2i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−2.23 + 0.133i)10-s + (2.5 + 4.33i)11-s + (−0.866 − 0.499i)12-s i·13-s + (−2.5 + 0.866i)14-s + (1 − 2i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.834 + 0.550i)5-s − 0.408·6-s + (−0.654 + 0.755i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.705 + 0.0423i)10-s + (0.753 + 1.30i)11-s + (−0.249 − 0.144i)12-s − 0.277i·13-s + (−0.668 + 0.231i)14-s + (0.258 − 0.516i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.441 - 0.897i$
Motivic weight: \(1\)
Character: $\chi_{210} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605913 + 0.972923i\)
\(L(\frac12)\) \(\approx\) \(0.605913 + 0.972923i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 + (1.73 - 2i)T \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.33 - 2.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (-11.2 - 6.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.19 + 3i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−12.42263408641630711772957514668, −11.93964500421718044762681284779, −11.00958781882818426725064451390, −9.791023116036347010493211691813, −8.720812962208541053074749127106, −7.15796884035051249280712395022, −6.67120912910708044868001964401, −5.26274379355380393494651105177, −4.16994536794863619644109203645, −2.87473553503323535748350655839, 0.919603030003091998088567393704, 3.40335185576002824914397478097, 4.32491177067244991324368399008, 5.74320376230786955381978504158, 6.74528461368143807122517886646, 7.896998875658963270868561172397, 9.179084383082505216020231608307, 10.40345061132740061453920542890, 11.45689796262071053221661627506, 11.91656303032767270314430561246

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