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Properties

Label 2-56-7.4-c3-0-2
Degree $2$
Conductor $56$
Sign $0.869 - 0.494i$
Analytic cond. $3.30410$
Root an. cond. $1.81772$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0691 + 0.119i)3-s + (5.04 + 8.73i)5-s + (18.4 − 1.05i)7-s + (13.4 + 23.3i)9-s + (9.33 − 16.1i)11-s + 10.0·13-s − 1.39·15-s + (−36.8 + 63.7i)17-s + (−38.0 − 65.8i)19-s + (−1.15 + 2.28i)21-s + (−73.4 − 127. i)23-s + (11.6 − 20.1i)25-s − 7.46·27-s − 157.·29-s + (−34.9 + 60.4i)31-s + ⋯
L(s)  = 1  + (−0.0132 + 0.0230i)3-s + (0.451 + 0.781i)5-s + (0.998 − 0.0567i)7-s + (0.499 + 0.865i)9-s + (0.255 − 0.443i)11-s + 0.215·13-s − 0.0239·15-s + (−0.525 + 0.909i)17-s + (−0.459 − 0.795i)19-s + (−0.0119 + 0.0237i)21-s + (−0.665 − 1.15i)23-s + (0.0931 − 0.161i)25-s − 0.0531·27-s − 1.00·29-s + (−0.202 + 0.350i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(3.30410\)
Root analytic conductor: \(1.81772\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :3/2),\ 0.869 - 0.494i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.54139 + 0.407542i\)
\(L(\frac12)\) \(\approx\) \(1.54139 + 0.407542i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-18.4 + 1.05i)T \)
good3 \( 1 + (0.0691 - 0.119i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-5.04 - 8.73i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-9.33 + 16.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 10.0T + 2.19e3T^{2} \)
17 \( 1 + (36.8 - 63.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (38.0 + 65.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (73.4 + 127. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 157.T + 2.43e4T^{2} \)
31 \( 1 + (34.9 - 60.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (154. + 268. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 482.T + 6.89e4T^{2} \)
43 \( 1 - 17.3T + 7.95e4T^{2} \)
47 \( 1 + (-173. - 300. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-36.7 + 63.6i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (352. - 610. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (420. + 728. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-445. + 772. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 525.T + 3.57e5T^{2} \)
73 \( 1 + (188. - 325. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (616. + 1.06e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 771.T + 5.71e5T^{2} \)
89 \( 1 + (-87.6 - 151. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.31e3T + 9.12e5T^{2} \)
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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

−14.65187352285414415441324821246, −13.93446112570549845507130669980, −12.70660083113231264456333147642, −10.98502865276577265670628646742, −10.61437520414087864058403516712, −8.831657745736344472981120551360, −7.53197760061148274445282167113, −6.09277128554813483199837231934, −4.40038799964263214820310515452, −2.12918730859827614386712297076, 1.54157623201242190477892817692, 4.23553283031908743570103503564, 5.66533613846618373688366137702, 7.35751192589140488286398480565, 8.826463145363937816124786875799, 9.785447113212345961083300406888, 11.38870459193209913053706833087, 12.39468794052016522420268386441, 13.51337710466603223866923623237, 14.69092306143577218847992898089

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