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Properties

Label 2-336-84.23-c3-0-32
Degree $2$
Conductor $336$
Sign $0.876 + 0.481i$
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  + (5.18 + 0.283i)3-s + (−8.90 − 5.14i)5-s + (−9.06 + 16.1i)7-s + (26.8 + 2.93i)9-s + (−23.0 − 40.0i)11-s + 26.5·13-s + (−44.7 − 29.2i)15-s + (102. − 59.4i)17-s + (142. + 82.5i)19-s + (−51.5 + 81.2i)21-s + (65.7 − 113. i)23-s + (−9.59 − 16.6i)25-s + (138. + 22.8i)27-s + 15.1i·29-s + (101. − 58.4i)31-s + ⋯
L(s)  = 1  + (0.998 + 0.0544i)3-s + (−0.796 − 0.460i)5-s + (−0.489 + 0.872i)7-s + (0.994 + 0.108i)9-s + (−0.633 − 1.09i)11-s + 0.566·13-s + (−0.770 − 0.502i)15-s + (1.46 − 0.848i)17-s + (1.72 + 0.996i)19-s + (−0.536 + 0.844i)21-s + (0.596 − 1.03i)23-s + (−0.0767 − 0.132i)25-s + (0.986 + 0.162i)27-s + 0.0968i·29-s + (0.586 − 0.338i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.876 + 0.481i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.876 + 0.481i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.320986088\)
\(L(\frac12)\) \(\approx\) \(2.320986088\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.18 - 0.283i)T \)
7 \( 1 + (9.06 - 16.1i)T \)
good5 \( 1 + (8.90 + 5.14i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (23.0 + 40.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 26.5T + 2.19e3T^{2} \)
17 \( 1 + (-102. + 59.4i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-142. - 82.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-65.7 + 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 15.1iT - 2.43e4T^{2} \)
31 \( 1 + (-101. + 58.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-113. + 196. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 220. iT - 6.89e4T^{2} \)
43 \( 1 - 338. iT - 7.95e4T^{2} \)
47 \( 1 + (169. - 293. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (660. - 381. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (238. + 412. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (148. - 257. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-596. + 344. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 700.T + 3.57e5T^{2} \)
73 \( 1 + (155. + 268. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-248. - 143. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 906.T + 5.71e5T^{2} \)
89 \( 1 + (616. + 355. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 354.T + 9.12e5T^{2} \)
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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

−11.10785409892155663151433365551, −9.834770834327414119606172585938, −9.138941740185693673753129780595, −8.076934178410728030619005295861, −7.75427251340680619808582182842, −6.10976873021255057791629218792, −5.00539430833186236533158707000, −3.50323148054737277886360119821, −2.88196947946648664379188744773, −0.906097643908712267939793792866, 1.24446169140523778117580739425, 3.10967943360119726926312973438, 3.66714508778959917280912098134, 5.01290022585799408447888569956, 6.80755772103065483116958237439, 7.52649911844118047712493482170, 8.058785742777019919748102972362, 9.567738019662187369390350855913, 10.02136022688147425994928181925, 11.12787803611651337985378027477

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