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Properties

Label 2-56-56.37-c3-0-6
Degree $2$
Conductor $56$
Sign $0.104 + 0.994i$
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  + (−2.70 − 0.819i)2-s + (−8.54 + 4.93i)3-s + (6.65 + 4.43i)4-s + (3.19 + 1.84i)5-s + (27.1 − 6.35i)6-s + (−18.3 − 2.28i)7-s + (−14.3 − 17.4i)8-s + (35.2 − 60.9i)9-s + (−7.13 − 7.61i)10-s + (28.5 − 16.4i)11-s + (−78.8 − 5.07i)12-s − 33.0i·13-s + (47.8 + 21.2i)14-s − 36.4·15-s + (24.6 + 59.0i)16-s + (26.6 + 46.2i)17-s + ⋯
L(s)  = 1  + (−0.957 − 0.289i)2-s + (−1.64 + 0.949i)3-s + (0.832 + 0.554i)4-s + (0.285 + 0.165i)5-s + (1.84 − 0.432i)6-s + (−0.992 − 0.123i)7-s + (−0.635 − 0.771i)8-s + (1.30 − 2.25i)9-s + (−0.225 − 0.240i)10-s + (0.782 − 0.451i)11-s + (−1.89 − 0.122i)12-s − 0.704i·13-s + (0.913 + 0.405i)14-s − 0.626·15-s + (0.384 + 0.922i)16-s + (0.380 + 0.659i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.104 + 0.994i$
Analytic conductor: \(3.30410\)
Root analytic conductor: \(1.81772\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :3/2),\ 0.104 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.225427 - 0.202932i\)
\(L(\frac12)\) \(\approx\) \(0.225427 - 0.202932i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.70 + 0.819i)T \)
7 \( 1 + (18.3 + 2.28i)T \)
good3 \( 1 + (8.54 - 4.93i)T + (13.5 - 23.3i)T^{2} \)
5 \( 1 + (-3.19 - 1.84i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (-28.5 + 16.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 33.0iT - 2.19e3T^{2} \)
17 \( 1 + (-26.6 - 46.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (44.3 + 25.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-53.8 + 93.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 7.61iT - 2.43e4T^{2} \)
31 \( 1 + (70.9 + 122. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (326. + 188. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 123.T + 6.89e4T^{2} \)
43 \( 1 - 35.6iT - 7.95e4T^{2} \)
47 \( 1 + (-106. + 184. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-396. + 228. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-70.4 + 40.6i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (438. + 252. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (171. - 99.1i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 474.T + 3.57e5T^{2} \)
73 \( 1 + (-337. - 585. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-489. + 847. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 956. iT - 5.71e5T^{2} \)
89 \( 1 + (512. - 888. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 76.6T + 9.12e5T^{2} \)
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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

−15.09140601013311169466927089795, −12.77788623136702507932556961104, −11.88311383674327301356601243116, −10.66612665529434969505268178180, −10.16718746275267033133156545569, −9.005387494553308716318605085568, −6.74587838784679713707367878837, −5.89510383200841257996269875628, −3.74077915843195954610131636156, −0.37824750320952361832080947525, 1.48577639387637544214727698095, 5.44793151025188811760219005016, 6.56085163442995046761030365631, 7.26168642335604768049640589211, 9.246902270721017101429967509985, 10.39764493332172290854519147990, 11.65362746095655200304883608679, 12.33640137232006527026118662732, 13.68614519954669990020535760135, 15.55834110048096833227156421682

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