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Properties

Label 2-336-7.2-c5-0-27
Degree $2$
Conductor $336$
Sign $0.259 + 0.965i$
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 + (15.9 − 27.6i)5-s + (−85.7 − 97.2i)7-s + (−40.5 + 70.1i)9-s + (130. + 225. i)11-s + 769.·13-s − 287.·15-s + (776. + 1.34e3i)17-s + (375. − 649. i)19-s + (−372. + 1.10e3i)21-s + (377. − 653. i)23-s + (1.05e3 + 1.82e3i)25-s + 729·27-s + 6.00e3·29-s + (−3.21e3 − 5.55e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.285 − 0.495i)5-s + (−0.661 − 0.750i)7-s + (−0.166 + 0.288i)9-s + (0.325 + 0.562i)11-s + 1.26·13-s − 0.330·15-s + (0.651 + 1.12i)17-s + (0.238 − 0.412i)19-s + (−0.184 + 0.547i)21-s + (0.148 − 0.257i)23-s + (0.336 + 0.582i)25-s + 0.192·27-s + 1.32·29-s + (−0.599 − 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.259 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.259 + 0.965i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.259 + 0.965i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.972346161\)
\(L(\frac12)\) \(\approx\) \(1.972346161\)
\(L(\frac{7}{2})\) 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 \( 1 + (85.7 + 97.2i)T \)
good5 \( 1 + (-15.9 + 27.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-130. - 225. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 769.T + 3.71e5T^{2} \)
17 \( 1 + (-776. - 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-375. + 649. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-377. + 653. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 6.00e3T + 2.05e7T^{2} \)
31 \( 1 + (3.21e3 + 5.55e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-2.38e3 + 4.13e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 5.42e3T + 1.15e8T^{2} \)
43 \( 1 - 1.18e4T + 1.47e8T^{2} \)
47 \( 1 + (8.71e3 - 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.88e4 + 3.26e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.10e4 - 1.91e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-4.08e3 + 7.07e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-6.50e3 - 1.12e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.23e4T + 1.80e9T^{2} \)
73 \( 1 + (2.18e4 + 3.77e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.83e4 + 6.64e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 2.18e4T + 3.93e9T^{2} \)
89 \( 1 + (-6.84e4 + 1.18e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 9.30e4T + 8.58e9T^{2} \)
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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

−10.56224726641397445752971538147, −9.634514542939558864015076525016, −8.643343104445163386981360272890, −7.60261354095062906589696762031, −6.57053559713282391449863277460, −5.83416751007355185983749496671, −4.47404019280169835613789429965, −3.35109684308924565263732609691, −1.62680222066596273511391430440, −0.69604791980659457119804459844, 0.985749477749371698055689997601, 2.80447059274564964898659831077, 3.59695772758583499943432022064, 5.14048641332467002720739199989, 6.06768312563463284023198047654, 6.77336161465408193794589479349, 8.311743213323337998461137933924, 9.156702124646531928278623723381, 10.00743931545747964328535568731, 10.86998597016009448676062059187

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