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Properties

Label 2-336-84.11-c3-0-34
Degree $2$
Conductor $336$
Sign $0.874 + 0.485i$
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  + (4.74 − 2.11i)3-s + (7.60 − 4.38i)5-s + (3.39 + 18.2i)7-s + (18.0 − 20.1i)9-s + (12.1 − 21.1i)11-s + 2.78·13-s + (26.7 − 36.9i)15-s + (23.4 + 13.5i)17-s + (57.2 − 33.0i)19-s + (54.6 + 79.1i)21-s + (44.5 + 77.1i)23-s + (−23.9 + 41.5i)25-s + (42.9 − 133. i)27-s − 77.3i·29-s + (76.0 + 43.9i)31-s + ⋯
L(s)  = 1  + (0.913 − 0.407i)3-s + (0.679 − 0.392i)5-s + (0.183 + 0.983i)7-s + (0.667 − 0.744i)9-s + (0.334 − 0.578i)11-s + 0.0594·13-s + (0.460 − 0.635i)15-s + (0.335 + 0.193i)17-s + (0.691 − 0.398i)19-s + (0.568 + 0.822i)21-s + (0.403 + 0.699i)23-s + (−0.191 + 0.332i)25-s + (0.306 − 0.951i)27-s − 0.495i·29-s + (0.440 + 0.254i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.874 + 0.485i$
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: \((2,\ 336,\ (\ :3/2),\ 0.874 + 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.142105384\)
\(L(\frac12)\) \(\approx\) \(3.142105384\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.74 + 2.11i)T \)
7 \( 1 + (-3.39 - 18.2i)T \)
good5 \( 1 + (-7.60 + 4.38i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-12.1 + 21.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.78T + 2.19e3T^{2} \)
17 \( 1 + (-23.4 - 13.5i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-57.2 + 33.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-44.5 - 77.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 77.3iT - 2.43e4T^{2} \)
31 \( 1 + (-76.0 - 43.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (107. + 186. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 197. iT - 6.89e4T^{2} \)
43 \( 1 + 251. iT - 7.95e4T^{2} \)
47 \( 1 + (-307. - 531. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (254. + 147. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-78.1 + 135. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (215. + 373. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-414. - 239. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 794.T + 3.57e5T^{2} \)
73 \( 1 + (372. - 644. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-389. + 224. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + (-578. + 334. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.67e3T + 9.12e5T^{2} \)
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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

−11.15266484121105444313554486242, −9.706911243081972807080067940616, −9.151997095778523192433658392369, −8.401653349465898101400975254765, −7.35975896337516149404628670301, −6.11907995697268902166002298171, −5.23064193891920844019136641861, −3.60336728494963973234153831453, −2.41990745420374873837726379204, −1.24800918365231408702222648900, 1.43466812888832217723971568140, 2.80178022554049193102876765962, 3.97608596189537600291996355646, 5.01617209712693155231928673865, 6.55621011987047402459741030417, 7.44292131077706368480098547365, 8.381186066477136797664120798032, 9.582443097143248372215621473977, 10.08796693904909322947707136068, 10.87518190808114659580684029855

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