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Properties

Label 2-336-7.4-c5-0-11
Degree $2$
Conductor $336$
Sign $0.997 + 0.0633i$
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 + (−23.0 − 39.9i)5-s + (−112. − 64.8i)7-s + (−40.5 − 70.1i)9-s + (−315. + 546. i)11-s − 1.07e3·13-s + 415.·15-s + (−80.5 + 139. i)17-s + (588. + 1.01e3i)19-s + (1.01e3 − 583. i)21-s + (1.08e3 + 1.87e3i)23-s + (499. − 864. i)25-s + 729·27-s − 4.49e3·29-s + (159. − 275. i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.412 − 0.714i)5-s + (−0.866 − 0.500i)7-s + (−0.166 − 0.288i)9-s + (−0.786 + 1.36i)11-s − 1.77·13-s + 0.476·15-s + (−0.0676 + 0.117i)17-s + (0.373 + 0.647i)19-s + (0.500 − 0.288i)21-s + (0.426 + 0.738i)23-s + (0.159 − 0.276i)25-s + 0.192·27-s − 0.991·29-s + (0.0297 − 0.0515i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.997 + 0.0633i$
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: \((2,\ 336,\ (\ :5/2),\ 0.997 + 0.0633i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7242016685\)
\(L(\frac12)\) \(\approx\) \(0.7242016685\)
\(L(\frac{7}{2})\) 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 \( 1 + (112. + 64.8i)T \)
good5 \( 1 + (23.0 + 39.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (315. - 546. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.07e3T + 3.71e5T^{2} \)
17 \( 1 + (80.5 - 139. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-588. - 1.01e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.08e3 - 1.87e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.49e3T + 2.05e7T^{2} \)
31 \( 1 + (-159. + 275. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (7.59e3 + 1.31e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 2.05e4T + 1.15e8T^{2} \)
43 \( 1 - 455.T + 1.47e8T^{2} \)
47 \( 1 + (1.03e4 + 1.79e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-9.65e3 + 1.67e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.18e3 + 5.51e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.45e4 - 4.25e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.70e4 + 2.94e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.29e4T + 1.80e9T^{2} \)
73 \( 1 + (-4.43e3 + 7.67e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.72e4 - 2.98e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.04e3T + 3.93e9T^{2} \)
89 \( 1 + (-1.01e4 - 1.75e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 5.40e4T + 8.58e9T^{2} \)
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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

−10.44517266378989046345682291355, −9.847912284549009397454384803374, −9.138789131904982818221168212422, −7.62467772802346415465593170838, −7.13455451830420657144858264259, −5.53093966484648346036401947415, −4.73033251472791356596653335256, −3.76193819262716732753414255289, −2.27253975207949117204532722995, −0.41461024099322164872782991885, 0.48241556074397316826224624743, 2.57475977501829034995939117137, 3.14347677707053215044989563635, 4.94421936784126188564644684462, 5.94781247935481404766554177003, 6.95597211716016772242961381488, 7.65982533439273066505778118539, 8.845514393271147714864238814473, 9.876716618885253766517915784446, 10.88949150775372175297741028733

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