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Properties

Label 2-336-7.4-c3-0-12
Degree $2$
Conductor $336$
Sign $0.805 - 0.592i$
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 + (10.6 + 18.5i)5-s + (18.2 − 3.19i)7-s + (−4.5 − 7.79i)9-s + (5.35 − 9.26i)11-s + 8.31·13-s + 64.1·15-s + (−0.742 + 1.28i)17-s + (80.7 + 139. i)19-s + (19.0 − 52.1i)21-s + (−61.9 − 107. i)23-s + (−166. + 288. i)25-s − 27·27-s − 222.·29-s + (101. − 176. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.956 + 1.65i)5-s + (0.984 − 0.172i)7-s + (−0.166 − 0.288i)9-s + (0.146 − 0.254i)11-s + 0.177·13-s + 1.10·15-s + (−0.0105 + 0.0183i)17-s + (0.975 + 1.68i)19-s + (0.198 − 0.542i)21-s + (−0.561 − 0.973i)23-s + (−1.33 + 2.30i)25-s − 0.192·27-s − 1.42·29-s + (0.590 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.805 - 0.592i$
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: \((2,\ 336,\ (\ :3/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.743865452\)
\(L(\frac12)\) \(\approx\) \(2.743865452\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.2 + 3.19i)T \)
good5 \( 1 + (-10.6 - 18.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-5.35 + 9.26i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 8.31T + 2.19e3T^{2} \)
17 \( 1 + (0.742 - 1.28i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-80.7 - 139. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (61.9 + 107. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 222.T + 2.43e4T^{2} \)
31 \( 1 + (-101. + 176. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-170. - 295. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 147.T + 6.89e4T^{2} \)
43 \( 1 + 51.3T + 7.95e4T^{2} \)
47 \( 1 + (-134. - 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-216. + 374. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-15.3 + 26.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-296. - 513. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-207. + 358. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 294.T + 3.57e5T^{2} \)
73 \( 1 + (-297. + 514. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (415. + 720. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 326.T + 5.71e5T^{2} \)
89 \( 1 + (-630. - 1.09e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−11.18420832575973208106880573114, −10.31170885796843161147328885469, −9.562222329040325283633462328100, −8.121909469567608363274176720481, −7.44827714318098950556414238758, −6.34667596239950781653967157028, −5.65442511572058073763705673474, −3.82171439316299136748098236939, −2.58732358850318283383138095580, −1.55645030340641848042154032163, 1.06659293848363275262715768338, 2.23755259585099508961440100353, 4.15990444283743466823064403846, 5.10796200301954845500892144145, 5.64638934689864623197982019901, 7.41901828105466205218190100505, 8.522937604884288189463361793219, 9.177879476577179061091301940139, 9.763147593807423001580504615618, 11.07783766040937718640314741264

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