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Properties

Label 2-336-84.11-c3-0-0
Degree $2$
Conductor $336$
Sign $-0.222 - 0.974i$
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  + (−0.538 − 5.16i)3-s + (−7.60 + 4.38i)5-s + (−3.39 − 18.2i)7-s + (−26.4 + 5.56i)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 + (−92.2 + 27.3i)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  + (−0.103 − 0.994i)3-s + (−0.679 + 0.392i)5-s + (−0.183 − 0.983i)7-s + (−0.978 + 0.205i)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 + (−0.958 + 0.284i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.222 - 0.974i$
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: \((2,\ 336,\ (\ :3/2),\ -0.222 - 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1660137054\)
\(L(\frac12)\) \(\approx\) \(0.1660137054\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.538 + 5.16i)T \)
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

−11.14357057080246378528839935628, −10.98771714465064675481678310796, −9.459150383453243695569559251541, −8.303926052224061865181012575122, −7.45505422168855510845764897967, −6.79826141246219784689502400463, −5.74367883291896755913873232003, −4.12869392254603368590403838666, −3.04711581996488050445348538227, −1.32530001217028331336656045826, 0.06256151076751042374064897157, 2.40438129259771219781543630435, 3.83451888705056759483073419949, 4.70466596229609862687284300594, 5.74271968874040412495572722955, 6.92517639907925235625980855369, 8.514504283528078213642500070169, 8.818009779093570602397991091932, 9.936240423310823938085155725779, 10.82247997907445084468861241992

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