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Properties

Label 2-336-7.2-c3-0-13
Degree $2$
Conductor $336$
Sign $0.850 + 0.526i$
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  + (−1.5 − 2.59i)3-s + (6.21 − 10.7i)5-s + (18.4 + 1.73i)7-s + (−4.5 + 7.79i)9-s + (30.1 + 52.2i)11-s + 36.4·13-s − 37.3·15-s + (24.3 + 42.2i)17-s + (−25.2 + 43.7i)19-s + (−23.1 − 50.5i)21-s + (69.3 − 120. i)23-s + (−14.8 − 25.6i)25-s + 27·27-s − 61.1·29-s + (−0.584 − 1.01i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.556 − 0.963i)5-s + (0.995 + 0.0938i)7-s + (−0.166 + 0.288i)9-s + (0.826 + 1.43i)11-s + 0.777·13-s − 0.642·15-s + (0.347 + 0.602i)17-s + (−0.305 + 0.528i)19-s + (−0.240 − 0.524i)21-s + (0.629 − 1.08i)23-s + (−0.118 − 0.205i)25-s + 0.192·27-s − 0.391·29-s + (−0.00338 − 0.00586i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.850 + 0.526i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.850 + 0.526i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.294208525\)
\(L(\frac12)\) \(\approx\) \(2.294208525\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (-18.4 - 1.73i)T \)
good5 \( 1 + (-6.21 + 10.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.1 - 52.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 + (-24.3 - 42.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (25.2 - 43.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-69.3 + 120. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 61.1T + 2.43e4T^{2} \)
31 \( 1 + (0.584 + 1.01i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (34.7 - 60.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 308.T + 6.89e4T^{2} \)
43 \( 1 + 174.T + 7.95e4T^{2} \)
47 \( 1 + (-194. + 337. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (157. + 272. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-422. - 731. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-169. + 293. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (485. + 841. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 98.4T + 3.57e5T^{2} \)
73 \( 1 + (355. + 615. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (243. - 421. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 605.T + 5.71e5T^{2} \)
89 \( 1 + (109. - 188. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 782.T + 9.12e5T^{2} \)
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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

−11.13995787620636941058802603667, −10.13338133084556852334096501088, −9.010820963341981334058845537921, −8.326512486830536381971115652517, −7.20097523294574123359944807103, −6.09692657588095644764165958567, −5.07946721374851089012198116396, −4.16926607497905593519412166345, −1.95340454578621793811353057383, −1.20296048583538723598368712157, 1.14645049960834012591034312431, 2.89157576292541202885501124459, 4.00975003014972250394910976009, 5.42464831209032576962435589618, 6.19388569528521067533303405279, 7.28200726202629948472836109769, 8.554928197684667162547025922578, 9.352228725499517382054131957358, 10.52361740492054247036455258308, 11.24834241535539509192691718361

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