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Properties

Label 2-336-28.19-c3-0-4
Degree $2$
Conductor $336$
Sign $0.113 - 0.993i$
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  + (−1.5 − 2.59i)3-s + (10.5 + 6.07i)5-s + (−16.1 − 9.05i)7-s + (−4.5 + 7.79i)9-s + (−0.668 + 0.385i)11-s + 61.4i·13-s − 36.4i·15-s + (16.0 − 9.27i)17-s + (−40.2 + 69.6i)19-s + (0.703 + 55.5i)21-s + (−4.96 − 2.86i)23-s + (11.2 + 19.4i)25-s + 27·27-s − 28.8·29-s + (55.7 + 96.5i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.940 + 0.543i)5-s + (−0.872 − 0.488i)7-s + (−0.166 + 0.288i)9-s + (−0.0183 + 0.0105i)11-s + 1.31i·13-s − 0.627i·15-s + (0.229 − 0.132i)17-s + (−0.485 + 0.840i)19-s + (0.00731 + 0.577i)21-s + (−0.0450 − 0.0259i)23-s + (0.0899 + 0.155i)25-s + 0.192·27-s − 0.184·29-s + (0.322 + 0.559i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.234633437\)
\(L(\frac12)\) \(\approx\) \(1.234633437\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (16.1 + 9.05i)T \)
good5 \( 1 + (-10.5 - 6.07i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (0.668 - 0.385i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 61.4iT - 2.19e3T^{2} \)
17 \( 1 + (-16.0 + 9.27i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40.2 - 69.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (4.96 + 2.86i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 28.8T + 2.43e4T^{2} \)
31 \( 1 + (-55.7 - 96.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (116. - 201. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 227. iT - 6.89e4T^{2} \)
43 \( 1 - 40.4iT - 7.95e4T^{2} \)
47 \( 1 + (74.8 - 129. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-340. - 588. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-171. - 297. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-292. - 169. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-730. + 421. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 912. iT - 3.57e5T^{2} \)
73 \( 1 + (256. - 148. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (415. + 239. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.21e3T + 5.71e5T^{2} \)
89 \( 1 + (866. + 500. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 258. iT - 9.12e5T^{2} \)
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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

−11.33979132391642071149234036858, −10.25155311028419532779478275880, −9.748524071496405444712913817956, −8.569284337708284992439745498113, −7.18323522608034698770557946931, −6.52938506194043468545229631496, −5.77614161165592615091169534918, −4.22381951635100313825536186168, −2.77647144706211117118419845431, −1.48289387703077033374632767525, 0.45227581939021660646635115046, 2.34772642135552704005659515055, 3.65657508912757267200719128800, 5.22904694376132813681463174187, 5.73181098132650321053852735103, 6.82942802085839098196079140138, 8.337122172377502797436145134510, 9.235226328112535567062670851225, 9.930255468673911020513954464963, 10.68751285948509436465549885159

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