Properties

Label 2-336-12.11-c3-0-19
Degree $2$
Conductor $336$
Sign $0.466 + 0.884i$
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  + (−4.59 + 2.42i)3-s + 2.41i·5-s + 7i·7-s + (15.2 − 22.2i)9-s − 37.7·11-s − 24.8·13-s + (−5.83 − 11.0i)15-s − 22.2i·17-s + 68.4i·19-s + (−16.9 − 32.1i)21-s − 44.4·23-s + 119.·25-s + (−16.2 + 139. i)27-s − 178. i·29-s − 109. i·31-s + ⋯
L(s)  = 1  + (−0.884 + 0.466i)3-s + 0.215i·5-s + 0.377i·7-s + (0.565 − 0.824i)9-s − 1.03·11-s − 0.529·13-s + (−0.100 − 0.190i)15-s − 0.317i·17-s + 0.826i·19-s + (−0.176 − 0.334i)21-s − 0.403·23-s + 0.953·25-s + (−0.116 + 0.993i)27-s − 1.14i·29-s − 0.636i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.466 + 0.884i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.466 + 0.884i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7401489284\)
\(L(\frac12)\) \(\approx\) \(0.7401489284\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.59 - 2.42i)T \)
7 \( 1 - 7iT \)
good5 \( 1 - 2.41iT - 125T^{2} \)
11 \( 1 + 37.7T + 1.33e3T^{2} \)
13 \( 1 + 24.8T + 2.19e3T^{2} \)
17 \( 1 + 22.2iT - 4.91e3T^{2} \)
19 \( 1 - 68.4iT - 6.85e3T^{2} \)
23 \( 1 + 44.4T + 1.21e4T^{2} \)
29 \( 1 + 178. iT - 2.43e4T^{2} \)
31 \( 1 + 109. iT - 2.97e4T^{2} \)
37 \( 1 - 168.T + 5.06e4T^{2} \)
41 \( 1 + 383. iT - 6.89e4T^{2} \)
43 \( 1 + 371. iT - 7.95e4T^{2} \)
47 \( 1 - 323.T + 1.03e5T^{2} \)
53 \( 1 + 401. iT - 1.48e5T^{2} \)
59 \( 1 - 34.0T + 2.05e5T^{2} \)
61 \( 1 - 25.4T + 2.26e5T^{2} \)
67 \( 1 - 118. iT - 3.00e5T^{2} \)
71 \( 1 - 106.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3T + 3.89e5T^{2} \)
79 \( 1 + 649. iT - 4.93e5T^{2} \)
83 \( 1 + 250.T + 5.71e5T^{2} \)
89 \( 1 + 4.02iT - 7.04e5T^{2} \)
97 \( 1 + 454.T + 9.12e5T^{2} \)
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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

−10.86538311243544291688056101861, −10.19120817805832744167353915363, −9.352612392590802727095601292171, −8.066699455208226930504750966213, −7.01942901159617954033934195857, −5.85963966416490293941682362380, −5.13368740011886426935399087314, −3.92093760213106499819544285067, −2.39893836580177432338743985545, −0.34547317786873830648959549179, 1.08602230662034672769811064608, 2.70449369670599026849114737275, 4.54895275449230744786170091566, 5.29366991859358663644506159146, 6.48792251886861889073098872686, 7.37612802735842116610081651945, 8.254225687752695130348031227648, 9.596417704151501572610900202641, 10.62170062377369790185128414058, 11.14007317035691097159761880780

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