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Properties

Label 2-210-7.4-c3-0-6
Degree $2$
Conductor $210$
Sign $0.664 - 0.747i$
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 + (14.5 + 11.4i)7-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (5 − 8.66i)10-s + (−21.0 + 36.5i)11-s + (6.00 + 10.3i)12-s − 67.1·13-s + (5.17 − 36.6i)14-s + 15.0·15-s + (−8 − 13.8i)16-s + (−51.4 + 89.0i)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 + (0.787 + 0.616i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.578 + 1.00i)11-s + (0.144 + 0.249i)12-s − 1.43·13-s + (0.0988 − 0.700i)14-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.733 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.664 - 0.747i$
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: \((2,\ 210,\ (\ :3/2),\ 0.664 - 0.747i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14534 + 0.513977i\)
\(L(\frac12)\) \(\approx\) \(1.14534 + 0.513977i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.5 - 11.4i)T \)
good11 \( 1 + (21.0 - 36.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 67.1T + 2.19e3T^{2} \)
17 \( 1 + (51.4 - 89.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-45.8 - 79.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-45.7 - 79.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 23.4T + 2.43e4T^{2} \)
31 \( 1 + (-121. + 210. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (33.2 + 57.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 296.T + 6.89e4T^{2} \)
43 \( 1 - 186.T + 7.95e4T^{2} \)
47 \( 1 + (44.6 + 77.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (144. - 250. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (406. - 703. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-163. - 283. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-328. + 569. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 622.T + 3.57e5T^{2} \)
73 \( 1 + (-433. + 751. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-364. - 631. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.13e3T + 5.71e5T^{2} \)
89 \( 1 + (-302. - 523. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 402.T + 9.12e5T^{2} \)
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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

−12.15743727913135456439198707176, −11.09907995154400286704022271228, −10.06566002215501675777998527526, −9.221502603802265518233770311752, −7.957100605783426680182495640876, −7.36642397125505176918646889983, −5.76720952491935567303092537101, −4.40544537623334601670972290640, −2.61293865470010310172286477694, −1.78043317063670195835673674745, 0.56793223213623270083074822972, 2.68897573478312415297766159538, 4.68426459321033210576275555923, 5.16813493198484130569745212607, 6.87680672368604905631711647785, 7.82475159170796428386663701099, 8.797118181393022745532504440776, 9.638832763909734091184908127359, 10.67958028756434509523787163659, 11.51997626102833679947965119616

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