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Properties

Label 2-336-7.4-c5-0-5
Degree $2$
Conductor $336$
Sign $0.566 - 0.824i$
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 + (−43.5 − 75.4i)5-s + (113. − 63.1i)7-s + (−40.5 − 70.1i)9-s + (−239. + 414. i)11-s − 436.·13-s − 784.·15-s + (−1.12e3 + 1.95e3i)17-s + (1.33e3 + 2.30e3i)19-s + (17.5 − 1.16e3i)21-s + (−1.28e3 − 2.22e3i)23-s + (−2.23e3 + 3.87e3i)25-s − 729·27-s − 708.·29-s + (2.75e3 − 4.77e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.779 − 1.35i)5-s + (0.873 − 0.486i)7-s + (−0.166 − 0.288i)9-s + (−0.595 + 1.03i)11-s − 0.715·13-s − 0.900·15-s + (−0.947 + 1.64i)17-s + (0.846 + 1.46i)19-s + (0.00869 − 0.577i)21-s + (−0.506 − 0.876i)23-s + (−0.715 + 1.23i)25-s − 0.192·27-s − 0.156·29-s + (0.514 − 0.891i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.566 - 0.824i$
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: \((2,\ 336,\ (\ :5/2),\ 0.566 - 0.824i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9546688379\)
\(L(\frac12)\) \(\approx\) \(0.9546688379\)
\(L(\frac{7}{2})\) 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 \( 1 + (-113. + 63.1i)T \)
good5 \( 1 + (43.5 + 75.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (239. - 414. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 436.T + 3.71e5T^{2} \)
17 \( 1 + (1.12e3 - 1.95e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.33e3 - 2.30e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.28e3 + 2.22e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 708.T + 2.05e7T^{2} \)
31 \( 1 + (-2.75e3 + 4.77e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-457. - 792. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 8.86e3T + 1.15e8T^{2} \)
43 \( 1 - 7.91e3T + 1.47e8T^{2} \)
47 \( 1 + (-9.78e3 - 1.69e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.12e3 - 7.14e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.60e4 - 2.78e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.42e4 + 2.46e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (4.84e3 - 8.38e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.85e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.55e3 + 2.69e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.39e3 + 5.88e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.44e4T + 3.93e9T^{2} \)
89 \( 1 + (1.37e4 + 2.37e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 9.12e4T + 8.58e9T^{2} \)
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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

−10.92299846606428395277247027670, −9.891742406217980150760886468046, −8.722792645390664185208747094604, −7.88356028576509286290507295513, −7.56069310037646964902944285878, −5.92278341416218101987498299709, −4.60952679652362608409409110881, −4.11385752058801293836282142541, −2.12315138158500352493412751362, −1.12532738823838754152894204957, 0.25558179867210561600686952156, 2.54801014024911472429720034308, 3.08847689588628885930750833384, 4.54353824797354508011661617341, 5.48350579051231646745773935847, 7.02842537434247408111462609577, 7.62571789911891943437198164673, 8.704163854006706908953855721076, 9.641518876180494197398130189314, 10.92660192003388049187957194974

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