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Properties

Label 2-336-21.17-c3-0-36
Degree $2$
Conductor $336$
Sign $0.216 + 0.976i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
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  + (3.06 + 4.19i)3-s + (0.746 + 1.29i)5-s + (−15.9 − 9.49i)7-s + (−8.26 + 25.7i)9-s + (−38.1 − 22.0i)11-s − 63.3i·13-s + (−3.14 + 7.08i)15-s + (55.7 − 96.5i)17-s + (109. − 63.3i)19-s + (−8.80 − 95.8i)21-s + (−176. + 102. i)23-s + (61.3 − 106. i)25-s + (−133. + 43.9i)27-s + 115. i·29-s + (74.9 + 43.2i)31-s + ⋯
L(s)  = 1  + (0.588 + 0.808i)3-s + (0.0667 + 0.115i)5-s + (−0.858 − 0.512i)7-s + (−0.306 + 0.951i)9-s + (−1.04 − 0.604i)11-s − 1.35i·13-s + (−0.0541 + 0.122i)15-s + (0.795 − 1.37i)17-s + (1.32 − 0.764i)19-s + (−0.0915 − 0.995i)21-s + (−1.60 + 0.925i)23-s + (0.491 − 0.850i)25-s + (−0.949 + 0.313i)27-s + 0.738i·29-s + (0.434 + 0.250i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.216 + 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.273577335\)
\(L(\frac12)\) \(\approx\) \(1.273577335\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-3.06 - 4.19i)T \)
7 \( 1 + (15.9 + 9.49i)T \)
good5 \( 1 + (-0.746 - 1.29i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (38.1 + 22.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 63.3iT - 2.19e3T^{2} \)
17 \( 1 + (-55.7 + 96.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-109. + 63.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (176. - 102. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 115. iT - 2.43e4T^{2} \)
31 \( 1 + (-74.9 - 43.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (53.8 + 93.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 336.T + 6.89e4T^{2} \)
43 \( 1 + 191.T + 7.95e4T^{2} \)
47 \( 1 + (158. + 274. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (321. + 185. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (187. - 325. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-38.7 + 22.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-207. + 359. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 676. iT - 3.57e5T^{2} \)
73 \( 1 + (587. + 339. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (289. + 500. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 253.T + 5.71e5T^{2} \)
89 \( 1 + (350. + 607. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 142. iT - 9.12e5T^{2} \)
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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

−10.55737966072955560934712680670, −10.06007039958261652018552850850, −9.277970589675814139532177914537, −8.011747031839378770613150922297, −7.39683555784266245745547461004, −5.76359084435643452199104004809, −4.94945131675901690685505693933, −3.32978229092419475440719826833, −2.89518417152855620619413701648, −0.40907029327309633324669663776, 1.60287598509206684341853843026, 2.75575650758897273652160918074, 4.00570271746642733872152878908, 5.70140966354892166529898760048, 6.49898625713397992114194782264, 7.63950641452570722577468447730, 8.356579011879402744198729796054, 9.530208234612803901692073231007, 10.06559957337196442604814498458, 11.63407982330026688897473489950

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