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Properties

Label 2-336-7.4-c3-0-15
Degree $2$
Conductor $336$
Sign $0.485 + 0.874i$
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 + (−0.0642 − 0.111i)5-s + (0.866 + 18.4i)7-s + (−4.5 − 7.79i)9-s + (27.0 − 46.7i)11-s − 50.2·13-s − 0.385·15-s + (65.7 − 113. i)17-s + (45.7 + 79.2i)19-s + (49.3 + 25.4i)21-s + (89.7 + 155. i)23-s + (62.4 − 108. i)25-s − 27·27-s − 69.8·29-s + (163. − 283. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.00575 − 0.00996i)5-s + (0.0467 + 0.998i)7-s + (−0.166 − 0.288i)9-s + (0.740 − 1.28i)11-s − 1.07·13-s − 0.00664·15-s + (0.937 − 1.62i)17-s + (0.552 + 0.956i)19-s + (0.512 + 0.264i)21-s + (0.813 + 1.40i)23-s + (0.499 − 0.865i)25-s − 0.192·27-s − 0.447·29-s + (0.946 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.025406355\)
\(L(\frac12)\) \(\approx\) \(2.025406355\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.866 - 18.4i)T \)
good5 \( 1 + (0.0642 + 0.111i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-27.0 + 46.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 50.2T + 2.19e3T^{2} \)
17 \( 1 + (-65.7 + 113. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-45.7 - 79.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-89.7 - 155. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 69.8T + 2.43e4T^{2} \)
31 \( 1 + (-163. + 283. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (150. + 261. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 296.T + 6.89e4T^{2} \)
43 \( 1 - 144.T + 7.95e4T^{2} \)
47 \( 1 + (180. + 311. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (0.917 - 1.58i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (26.6 - 46.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-54.0 - 93.6i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-421. + 729. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 241.T + 3.57e5T^{2} \)
73 \( 1 + (-103. + 179. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-279. - 484. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 986.T + 5.71e5T^{2} \)
89 \( 1 + (-221. - 383. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 740.T + 9.12e5T^{2} \)
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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

−11.33092736841616967924999644320, −9.689663101646931779734534805799, −9.204376680301887323530008688352, −8.070527783082231358381892762878, −7.27184372070020754936218935828, −5.98674079534742615482379063908, −5.22600964195533709368425776442, −3.46618836089878856741497388621, −2.44533707818752122913459930794, −0.794934256399545305208804907983, 1.32599867473000601955002908531, 3.01176139457744335426120800341, 4.28134194355313791893139049414, 4.99141411179762815320219034927, 6.70153255079097296054252728568, 7.37505220604542685389933494302, 8.528294152787050997554307563811, 9.624518599272520102165114710146, 10.23084981457941925747122610913, 11.06902197303340073151544809636

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