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Properties

Label 2-336-7.4-c5-0-30
Degree $2$
Conductor $336$
Sign $-0.415 + 0.909i$
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  + (−4.5 + 7.79i)3-s + (−39.3 − 68.1i)5-s + (100. − 81.7i)7-s + (−40.5 − 70.1i)9-s + (345. − 598. i)11-s + 818.·13-s + 708.·15-s + (−554. + 960. i)17-s + (−286. − 496. i)19-s + (184. + 1.15e3i)21-s + (1.25e3 + 2.18e3i)23-s + (−1.53e3 + 2.65e3i)25-s + 729·27-s − 3.25e3·29-s + (5.05e3 − 8.76e3i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.703 − 1.21i)5-s + (0.776 − 0.630i)7-s + (−0.166 − 0.288i)9-s + (0.861 − 1.49i)11-s + 1.34·13-s + 0.812·15-s + (−0.465 + 0.806i)17-s + (−0.182 − 0.315i)19-s + (0.0913 + 0.570i)21-s + (0.496 + 0.859i)23-s + (−0.490 + 0.849i)25-s + 0.192·27-s − 0.719·29-s + (0.945 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.415 + 0.909i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.415 + 0.909i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.694919247\)
\(L(\frac12)\) \(\approx\) \(1.694919247\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 + (-100. + 81.7i)T \)
good5 \( 1 + (39.3 + 68.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-345. + 598. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 818.T + 3.71e5T^{2} \)
17 \( 1 + (554. - 960. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (286. + 496. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.25e3 - 2.18e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 3.25e3T + 2.05e7T^{2} \)
31 \( 1 + (-5.05e3 + 8.76e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.43e3 + 4.21e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.30e4T + 1.15e8T^{2} \)
43 \( 1 - 9.30e3T + 1.47e8T^{2} \)
47 \( 1 + (-6.45e3 - 1.11e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.77e3 + 1.69e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.25e4 - 2.17e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.56e4 - 2.71e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.79e4 + 4.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.05e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.38e4 + 5.85e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-7.03e3 - 1.21e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.71e4T + 3.93e9T^{2} \)
89 \( 1 + (160. + 277. i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.12e5T + 8.58e9T^{2} \)
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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

−10.79897528159520708500481927960, −9.222210736825257405929583888730, −8.591633277047178047575637201965, −7.906390793181312893657673089388, −6.34940505500678829598317717792, −5.38471617345523453743318496639, −4.15809551336087158537522234515, −3.73045858712527615340021456434, −1.32325323631302134976510551403, −0.52899115357372769023088824992, 1.37157113465126250620831576564, 2.55539323842507604912357399320, 3.91102396295932534384881139492, 5.07945929627687069850658830534, 6.58577783642857198413700431986, 6.95323723450063515447157222117, 8.092969212818592495838912274663, 8.998707562830886770489104047354, 10.37879396586550130175633964077, 11.16703463546694725980223926025

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