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Properties

Label 2-336-7.4-c3-0-21
Degree $2$
Conductor $336$
Sign $-0.949 + 0.314i$
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 + (−10.4 − 18.0i)5-s + (18.3 + 2.59i)7-s + (−4.5 − 7.79i)9-s + (7.58 − 13.1i)11-s + 2.16·13-s − 62.5·15-s + (59.6 − 103. i)17-s + (−16.7 − 29.0i)19-s + (34.2 − 43.7i)21-s + (0.325 + 0.564i)23-s + (−154. + 267. i)25-s − 27·27-s − 163.·29-s + (−111. + 193. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.931 − 1.61i)5-s + (0.990 + 0.140i)7-s + (−0.166 − 0.288i)9-s + (0.207 − 0.359i)11-s + 0.0461·13-s − 1.07·15-s + (0.851 − 1.47i)17-s + (−0.202 − 0.350i)19-s + (0.355 − 0.454i)21-s + (0.00295 + 0.00511i)23-s + (−1.23 + 2.14i)25-s − 0.192·27-s − 1.04·29-s + (−0.646 + 1.12i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.949 + 0.314i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.463294362\)
\(L(\frac12)\) \(\approx\) \(1.463294362\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 - 2.59i)T \)
good5 \( 1 + (10.4 + 18.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-7.58 + 13.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.16T + 2.19e3T^{2} \)
17 \( 1 + (-59.6 + 103. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (16.7 + 29.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-0.325 - 0.564i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 163.T + 2.43e4T^{2} \)
31 \( 1 + (111. - 193. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (84.2 + 145. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 323.T + 6.89e4T^{2} \)
43 \( 1 + 221.T + 7.95e4T^{2} \)
47 \( 1 + (-254. - 439. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-88.2 + 152. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-227. + 393. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (19.3 + 33.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-70.8 + 122. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 602.T + 3.57e5T^{2} \)
73 \( 1 + (-551. + 954. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (58.1 + 100. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 568.T + 5.71e5T^{2} \)
89 \( 1 + (-191. - 331. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 334.T + 9.12e5T^{2} \)
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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

−11.04725373652322976491645423977, −9.359305514025400964778163763951, −8.729236607109870570007885054485, −7.946152942935418293838386317322, −7.21973000688761068561840026897, −5.43800432911910498537792270691, −4.77683645063327750899184015223, −3.49570239090395751747798361021, −1.62407672577037477706632327052, −0.51553774487952311241337788124, 2.01258290574865809371375266234, 3.50910206829972718130434729273, 4.11697109124252165276177600566, 5.66357500489084533147386898518, 6.95635872653857160564095626899, 7.78352013800705380088756847472, 8.487624649907752353040796769109, 10.04860636071332438859874432155, 10.58794688413051497069161231522, 11.39739820412636315162371942375

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