Properties

Label 2-336-21.5-c3-0-26
Degree $2$
Conductor $336$
Sign $0.572 + 0.819i$
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  + (−2.59 + 4.50i)3-s + (−8.05 + 13.9i)5-s + (5.67 − 17.6i)7-s + (−13.5 − 23.3i)9-s + (−30.8 + 17.7i)11-s + 7.40i·13-s + (−41.9 − 72.4i)15-s + (−14.4 − 25.0i)17-s + (−30.4 − 17.5i)19-s + (64.6 + 71.2i)21-s + (−48.0 − 27.7i)23-s + (−67.3 − 116. i)25-s + (140. − 0.420i)27-s + 68.1i·29-s + (154. − 89.3i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.720 + 1.24i)5-s + (0.306 − 0.951i)7-s + (−0.501 − 0.865i)9-s + (−0.845 + 0.487i)11-s + 0.158i·13-s + (−0.722 − 1.24i)15-s + (−0.206 − 0.357i)17-s + (−0.367 − 0.212i)19-s + (0.671 + 0.740i)21-s + (−0.435 − 0.251i)23-s + (−0.539 − 0.933i)25-s + (0.999 − 0.00299i)27-s + 0.436i·29-s + (0.896 − 0.517i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.572 + 0.819i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.572 + 0.819i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5437027457\)
\(L(\frac12)\) \(\approx\) \(0.5437027457\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.59 - 4.50i)T \)
7 \( 1 + (-5.67 + 17.6i)T \)
good5 \( 1 + (8.05 - 13.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.8 - 17.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 7.40iT - 2.19e3T^{2} \)
17 \( 1 + (14.4 + 25.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (30.4 + 17.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (48.0 + 27.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 68.1iT - 2.43e4T^{2} \)
31 \( 1 + (-154. + 89.3i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-116. + 202. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 370.T + 6.89e4T^{2} \)
43 \( 1 - 187.T + 7.95e4T^{2} \)
47 \( 1 + (-87.3 + 151. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (235. - 136. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-48.4 - 83.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (333. + 192. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (509. + 881. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 125. iT - 3.57e5T^{2} \)
73 \( 1 + (-195. + 112. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (532. - 921. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 601.T + 5.71e5T^{2} \)
89 \( 1 + (-752. + 1.30e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 327. iT - 9.12e5T^{2} \)
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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

−10.81672337969843148777431261605, −10.43340281787721103595026281928, −9.387608267902169869115791912616, −7.922718558539933115898095616571, −7.17420137554534778179175768471, −6.15843957277302318441424478450, −4.70711561491265161870557585331, −3.93932131854150714319198897901, −2.72260839566187879189768621720, −0.23946650388903427875713569665, 1.10741071718529465301268641954, 2.57680342924427340015158676599, 4.46515249260122874435137511738, 5.40929946866665191575599510418, 6.22258868273828360190917939001, 7.84352123323279613344374986946, 8.179480859948325930206712942132, 9.094488456862205311867572771345, 10.59446691118417452315217830460, 11.57369170985079003161599040166

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