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Properties

Label 2-210-7.2-c7-0-5
Degree $2$
Conductor $210$
Sign $-0.588 - 0.808i$
Analytic cond. $65.6008$
Root an. cond. $8.09943$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−13.5 − 23.3i)3-s + (−31.9 − 55.4i)4-s + (62.5 − 108. i)5-s + 216·6-s + (765. − 486. i)7-s + 511.·8-s + (−364.5 + 631. i)9-s + (499. + 866. i)10-s + (3.91e3 + 6.77e3i)11-s + (−864. + 1.49e3i)12-s − 1.37e4·13-s + (307. + 7.25e3i)14-s − 3.37e3·15-s + (−2.04e3 + 3.54e3i)16-s + (−1.28e4 − 2.22e4i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s + (0.844 − 0.536i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.886 + 1.53i)11-s + (−0.144 + 0.249i)12-s − 1.73·13-s + (0.0299 + 0.706i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (−0.634 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(65.6008\)
Root analytic conductor: \(8.09943\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :7/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6853140537\)
\(L(\frac12)\) \(\approx\) \(0.6853140537\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
3 \( 1 + (13.5 + 23.3i)T \)
5 \( 1 + (-62.5 + 108. i)T \)
7 \( 1 + (-765. + 486. i)T \)
good11 \( 1 + (-3.91e3 - 6.77e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + 1.37e4T + 6.27e7T^{2} \)
17 \( 1 + (1.28e4 + 2.22e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (628. - 1.08e3i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (-5.36e3 + 9.29e3i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 - 7.50e4T + 1.72e10T^{2} \)
31 \( 1 + (-7.46e4 - 1.29e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (8.28e4 - 1.43e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 2.77e3T + 1.94e11T^{2} \)
43 \( 1 + 7.73e5T + 2.71e11T^{2} \)
47 \( 1 + (3.64e5 - 6.30e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (8.10e5 + 1.40e6i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-1.33e6 - 2.31e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-8.04e5 + 1.39e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.20e6 - 2.07e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + 5.04e6T + 9.09e12T^{2} \)
73 \( 1 + (1.38e3 + 2.39e3i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (1.42e6 - 2.47e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 7.06e6T + 2.71e13T^{2} \)
89 \( 1 + (-1.72e5 + 2.99e5i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.16e7T + 8.07e13T^{2} \)
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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

−11.62554515313652944049839242333, −10.18107118695777026549161306218, −9.496394799813366375608378379579, −8.282932193394317571149144763750, −7.18177238079959494628062817020, −6.79353791568557025089590029954, −4.97860550410443354553937435099, −4.64489517391602314203676362062, −2.20513815488988655206522057914, −1.15200108588981229292989999508, 0.20732614257625002937590611530, 1.74348728013400391263188772235, 2.95398341007060903414444177766, 4.24002524823097139830493390363, 5.40905717134839489466483942528, 6.57685585401877903540421326860, 8.077718916332658570302074165578, 8.907595948037511501196938814586, 9.878804091799714421308526955710, 10.84962756177122025849870014844

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