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Properties

Label 2-210-105.38-c2-0-13
Degree $2$
Conductor $210$
Sign $0.851 + 0.523i$
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  + (0.366 − 1.36i)2-s + (−1.90 + 2.32i)3-s + (−1.73 − i)4-s + (−4.63 − 1.87i)5-s + (2.47 + 3.44i)6-s + (6.54 + 2.48i)7-s + (−2 + 1.99i)8-s + (−1.77 − 8.82i)9-s + (−4.25 + 5.64i)10-s + (16.0 + 9.29i)11-s + (5.61 − 2.12i)12-s + (11.8 − 11.8i)13-s + (5.79 − 8.02i)14-s + (13.1 − 7.20i)15-s + (1.99 + 3.46i)16-s + (−4.38 − 16.3i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.633 + 0.773i)3-s + (−0.433 − 0.250i)4-s + (−0.927 − 0.374i)5-s + (0.412 + 0.574i)6-s + (0.934 + 0.355i)7-s + (−0.250 + 0.249i)8-s + (−0.197 − 0.980i)9-s + (−0.425 + 0.564i)10-s + (1.46 + 0.844i)11-s + (0.467 − 0.176i)12-s + (0.912 − 0.912i)13-s + (0.413 − 0.573i)14-s + (0.877 − 0.480i)15-s + (0.124 + 0.216i)16-s + (−0.257 − 0.962i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.851 + 0.523i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ 0.851 + 0.523i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.27146 - 0.359526i\)
\(L(\frac12)\) \(\approx\) \(1.27146 - 0.359526i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
3 \( 1 + (1.90 - 2.32i)T \)
5 \( 1 + (4.63 + 1.87i)T \)
7 \( 1 + (-6.54 - 2.48i)T \)
good11 \( 1 + (-16.0 - 9.29i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11.8 + 11.8i)T - 169iT^{2} \)
17 \( 1 + (4.38 + 16.3i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-2.06 - 3.58i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.3 - 5.44i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 49.5T + 841T^{2} \)
31 \( 1 + (2.73 + 1.57i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (30.3 + 8.12i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 26.6T + 1.68e3T^{2} \)
43 \( 1 + (25.3 + 25.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (49.0 + 13.1i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-10.1 - 37.7i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-46.6 - 26.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-34.2 + 19.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12.7 + 47.4i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 81.2iT - 5.04e3T^{2} \)
73 \( 1 + (0.107 + 0.400i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-117. + 67.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-27.5 - 27.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (20.2 - 11.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (50.1 + 50.1i)T + 9.40e3iT^{2} \)
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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

−11.80745783173411887311447106494, −11.35675936833500788737816329294, −10.35054494213011004317934452472, −9.161948627996029185759472346806, −8.424098070312863419854225445295, −6.83009429789620339467390048122, −5.25478234832988690128100837955, −4.50093760158341776362342484161, −3.43484665394111924351379321810, −1.07141541268569644797774459653, 1.19411336846668489662696379901, 3.73428313851122769513495767675, 4.83660333884312532761625899093, 6.47214840061828592542170105723, 6.78709452764183812861346253360, 8.191947370629056315949842828761, 8.655709594178153444876918068855, 10.70801258156294513945831898381, 11.46829480502047968877073626118, 12.00447160691614633958869356043

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