Properties

Label 2-336-12.11-c5-0-21
Degree $2$
Conductor $336$
Sign $0.390 - 0.920i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−14.3 − 6.08i)3-s + 63.6i·5-s + 49i·7-s + (169. + 174. i)9-s − 66.5·11-s + 1.15e3·13-s + (387. − 913. i)15-s − 1.56e3i·17-s − 191. i·19-s + (297. − 703. i)21-s − 2.35e3·23-s − 927.·25-s + (−1.36e3 − 3.53e3i)27-s + 3.55e3i·29-s + 3.81e3i·31-s + ⋯
L(s)  = 1  + (−0.920 − 0.390i)3-s + 1.13i·5-s + 0.377i·7-s + (0.695 + 0.718i)9-s − 0.165·11-s + 1.89·13-s + (0.444 − 1.04i)15-s − 1.31i·17-s − 0.121i·19-s + (0.147 − 0.348i)21-s − 0.926·23-s − 0.296·25-s + (−0.360 − 0.932i)27-s + 0.785i·29-s + 0.712i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.390 - 0.920i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.390 - 0.920i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.485804994\)
\(L(\frac12)\) \(\approx\) \(1.485804994\)
\(L(\frac{7}{2})\) 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 \( 1 + (14.3 + 6.08i)T \)
7 \( 1 - 49iT \)
good5 \( 1 - 63.6iT - 3.12e3T^{2} \)
11 \( 1 + 66.5T + 1.61e5T^{2} \)
13 \( 1 - 1.15e3T + 3.71e5T^{2} \)
17 \( 1 + 1.56e3iT - 1.41e6T^{2} \)
19 \( 1 + 191. iT - 2.47e6T^{2} \)
23 \( 1 + 2.35e3T + 6.43e6T^{2} \)
29 \( 1 - 3.55e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.81e3iT - 2.86e7T^{2} \)
37 \( 1 - 7.55e3T + 6.93e7T^{2} \)
41 \( 1 - 7.56e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.93e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.55e4T + 2.29e8T^{2} \)
53 \( 1 + 868. iT - 4.18e8T^{2} \)
59 \( 1 + 1.53e4T + 7.14e8T^{2} \)
61 \( 1 + 1.33e4T + 8.44e8T^{2} \)
67 \( 1 + 1.63e3iT - 1.35e9T^{2} \)
71 \( 1 - 7.38e4T + 1.80e9T^{2} \)
73 \( 1 - 3.27e4T + 2.07e9T^{2} \)
79 \( 1 - 5.34e4iT - 3.07e9T^{2} \)
83 \( 1 + 4.11e3T + 3.93e9T^{2} \)
89 \( 1 - 5.40e4iT - 5.58e9T^{2} \)
97 \( 1 + 2.05e4T + 8.58e9T^{2} \)
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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

−10.95895993809912607735451322113, −10.34788487321360412528598295602, −9.059122278572235571283124162890, −7.84232443825944807807236233148, −6.84188150982077173094530780312, −6.18324665240184206646596264022, −5.20579931443414417940541337511, −3.71726904867354289328209385604, −2.42811699083461510180302704418, −0.968567461955962914995757743523, 0.58137921268403749650624262856, 1.50579663452262602182048431649, 3.82554284460635459493457653269, 4.42308390459981755453400575791, 5.78278393984058098056195961406, 6.23780893247043941836558428345, 7.86153229474160476243782828768, 8.702519506311648952977308298966, 9.707267006815236076303691163516, 10.68417409589532851047391746256

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