Properties

Label 2-210-7.4-c3-0-11
Degree $2$
Conductor $210$
Sign $0.994 + 0.106i$
Analytic cond. $12.3904$
Root an. cond. $3.52000$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (2.5 + 4.33i)5-s + 6·6-s + (0.809 − 18.5i)7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−5 + 8.66i)10-s + (21.9 − 37.9i)11-s + (6.00 + 10.3i)12-s + 66.8·13-s + (32.8 − 17.1i)14-s + 15.0·15-s + (−8 − 13.8i)16-s + (−8.07 + 13.9i)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 + (0.0437 − 0.999i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.601 − 1.04i)11-s + (0.144 + 0.249i)12-s + 1.42·13-s + (0.627 − 0.326i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.115 + 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.994 + 0.106i$
Analytic conductor: \(12.3904\)
Root analytic conductor: \(3.52000\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :3/2),\ 0.994 + 0.106i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.46776 - 0.132271i\)
\(L(\frac12)\) \(\approx\) \(2.46776 - 0.132271i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (-1.5 + 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-0.809 + 18.5i)T \)
good11 \( 1 + (-21.9 + 37.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 66.8T + 2.19e3T^{2} \)
17 \( 1 + (8.07 - 13.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-38.7 - 67.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (51.1 + 88.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 192.T + 2.43e4T^{2} \)
31 \( 1 + (-151. + 263. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-183. - 318. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 7.85T + 6.89e4T^{2} \)
43 \( 1 + 182.T + 7.95e4T^{2} \)
47 \( 1 + (134. + 232. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (194. - 336. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (325. - 564. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (62.4 + 108. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-372. + 645. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 561.T + 3.57e5T^{2} \)
73 \( 1 + (-203. + 351. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-463. - 802. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.10e3T + 5.71e5T^{2} \)
89 \( 1 + (354. + 614. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 48.7T + 9.12e5T^{2} \)
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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

−11.97310793418517651629693818700, −11.03179023696734559643749796190, −9.926592051407334494780842827901, −8.512717869820626408984389881979, −7.899960061065117581749789744864, −6.50338972284633379612834440254, −6.11424098561685173778580675136, −4.22515567297225110421096963391, −3.19641210301668849153500188750, −1.07968726238013717621673585583, 1.54974713627413221485302555176, 2.98626283765743198602633853329, 4.34212500033205738181498333907, 5.34658872254876761753605079223, 6.54904105253620163509834851718, 8.332779748308530954963856929002, 9.169571783698582260179572024801, 9.868811593413914835794482253898, 11.11058759692492252088446510359, 11.92383988813770411620319248971

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