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Properties

Label 2-336-84.11-c3-0-46
Degree $2$
Conductor $336$
Sign $-0.794 + 0.606i$
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  + (0.617 − 5.15i)3-s + (17.2 − 9.94i)5-s + (−5.66 − 17.6i)7-s + (−26.2 − 6.37i)9-s + (22.0 − 38.2i)11-s − 32.6·13-s + (−40.6 − 95.0i)15-s + (78.8 + 45.5i)17-s + (−49.6 + 28.6i)19-s + (−94.4 + 18.3i)21-s + (−41.9 − 72.7i)23-s + (135. − 234. i)25-s + (−49.1 + 131. i)27-s + 188. i·29-s + (225. + 130. i)31-s + ⋯
L(s)  = 1  + (0.118 − 0.992i)3-s + (1.54 − 0.889i)5-s + (−0.305 − 0.952i)7-s + (−0.971 − 0.236i)9-s + (0.604 − 1.04i)11-s − 0.696·13-s + (−0.699 − 1.63i)15-s + (1.12 + 0.649i)17-s + (−0.599 + 0.345i)19-s + (−0.981 + 0.190i)21-s + (−0.380 − 0.659i)23-s + (1.08 − 1.87i)25-s + (−0.349 + 0.936i)27-s + 1.20i·29-s + (1.30 + 0.755i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.794 + 0.606i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.794 + 0.606i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.281550112\)
\(L(\frac12)\) \(\approx\) \(2.281550112\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.617 + 5.15i)T \)
7 \( 1 + (5.66 + 17.6i)T \)
good5 \( 1 + (-17.2 + 9.94i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-22.0 + 38.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 32.6T + 2.19e3T^{2} \)
17 \( 1 + (-78.8 - 45.5i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (49.6 - 28.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (41.9 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 188. iT - 2.43e4T^{2} \)
31 \( 1 + (-225. - 130. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-172. - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 34.4iT - 6.89e4T^{2} \)
43 \( 1 + 201. iT - 7.95e4T^{2} \)
47 \( 1 + (232. + 402. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (176. + 102. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (113. - 197. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (423. + 732. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (146. + 84.6i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 86.4T + 3.57e5T^{2} \)
73 \( 1 + (112. - 194. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-513. + 296. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 127.T + 5.71e5T^{2} \)
89 \( 1 + (-560. + 323. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 704.T + 9.12e5T^{2} \)
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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

−10.56140179044002827963938206982, −9.845616186246321723244886328340, −8.770832502649106871072275987599, −8.046816078316687600363844756183, −6.60583367390343374385979056371, −6.10525439773411232151750548948, −4.94121623183545556215097510993, −3.22618708256645376828246972104, −1.69795081463959382151358432382, −0.806993896532239534090321731336, 2.18244482093343687103554584085, 2.92307601131341989610898362279, 4.55917606293374280974222236768, 5.70324818097749315012778468911, 6.32614783243453233551250310388, 7.69980855833106610558985723245, 9.347069035217796304468263988498, 9.575885408698685476841983576383, 10.19698104343337272459192132186, 11.37603583303898310971687001967

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