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Properties

Label 2-210-7.4-c7-0-6
Degree $2$
Conductor $210$
Sign $0.518 - 0.855i$
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 + (−804. − 419. i)7-s + 511.·8-s + (−364.5 − 631. i)9-s + (499. − 866. i)10-s + (−1.24e3 + 2.16e3i)11-s + (−864. − 1.49e3i)12-s + 6.88e3·13-s + (308. + 7.25e3i)14-s − 3.37e3·15-s + (−2.04e3 − 3.54e3i)16-s + (1.95e3 − 3.38e3i)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 + (−0.886 − 0.462i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.282 + 0.489i)11-s + (−0.144 − 0.249i)12-s + 0.869·13-s + (0.0299 + 0.706i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (0.0966 − 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.518 - 0.855i$
Analytic conductor: \(65.6008\)
Root analytic conductor: \(8.09943\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :7/2),\ 0.518 - 0.855i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.9600689130\)
\(L(\frac12)\) \(\approx\) \(0.9600689130\)
\(L(\frac{9}{2})\) 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 \( 1 + (804. + 419. i)T \)
good11 \( 1 + (1.24e3 - 2.16e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 6.88e3T + 6.27e7T^{2} \)
17 \( 1 + (-1.95e3 + 3.38e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.17e4 + 3.76e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-1.77e4 - 3.06e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 1.65e5T + 1.72e10T^{2} \)
31 \( 1 + (-9.63e4 + 1.66e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (4.65e4 + 8.06e4i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 3.18e5T + 1.94e11T^{2} \)
43 \( 1 + 2.16e5T + 2.71e11T^{2} \)
47 \( 1 + (4.97e5 + 8.61e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (6.87e5 - 1.19e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (9.07e5 - 1.57e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.15e6 - 2.00e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-8.54e5 + 1.48e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 1.24e6T + 9.09e12T^{2} \)
73 \( 1 + (2.05e6 - 3.56e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-2.30e6 - 3.99e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 8.47e6T + 2.71e13T^{2} \)
89 \( 1 + (-2.76e6 - 4.79e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 8.84e5T + 8.07e13T^{2} \)
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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

−11.01992480202942487533402018118, −10.30983512397679021721205527408, −9.574059551923907517457282968131, −8.580619449083570005211640817509, −7.14776212368631944717541650539, −6.21544330151626586319390342979, −4.71973274934370835972365290070, −3.58464890625676372936482979372, −2.52792913786362986094220346272, −0.845507230003732514004737597581, 0.37355626197340142012459877382, 1.62945905567107324099756295514, 3.26758239754677200868612978508, 4.94778298385283654416789831490, 6.15890219186583834970957317050, 6.51991383298462256255925604024, 8.139304516974979002996011087175, 8.653617467623495296949202762445, 9.894739450323503657301894571079, 10.75012079904659736709591543749

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