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Properties

Label 2-56-7.4-c3-0-0
Degree $2$
Conductor $56$
Sign $-0.986 + 0.165i$
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  + (−4.25 + 7.37i)3-s + (−4.23 − 7.34i)5-s + (−17.7 − 5.36i)7-s + (−22.7 − 39.3i)9-s + (−12.5 + 21.6i)11-s − 21.5·13-s + 72.1·15-s + (−50.8 + 88.0i)17-s + (72.9 + 126. i)19-s + (114. − 107. i)21-s + (−88.9 − 154. i)23-s + (26.5 − 46.0i)25-s + 157.·27-s + 89.5·29-s + (−105. + 183. i)31-s + ⋯
L(s)  = 1  + (−0.819 + 1.41i)3-s + (−0.379 − 0.656i)5-s + (−0.957 − 0.289i)7-s + (−0.841 − 1.45i)9-s + (−0.342 + 0.593i)11-s − 0.460·13-s + 1.24·15-s + (−0.725 + 1.25i)17-s + (0.881 + 1.52i)19-s + (1.19 − 1.12i)21-s + (−0.806 − 1.39i)23-s + (0.212 − 0.368i)25-s + 1.11·27-s + 0.573·29-s + (−0.613 + 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.986 + 0.165i$
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: \((2,\ 56,\ (\ :3/2),\ -0.986 + 0.165i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0251665 - 0.301509i\)
\(L(\frac12)\) \(\approx\) \(0.0251665 - 0.301509i\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.7 + 5.36i)T \)
good3 \( 1 + (4.25 - 7.37i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (4.23 + 7.34i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (12.5 - 21.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 21.5T + 2.19e3T^{2} \)
17 \( 1 + (50.8 - 88.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-72.9 - 126. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (88.9 + 154. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 89.5T + 2.43e4T^{2} \)
31 \( 1 + (105. - 183. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-136. - 235. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 50.4T + 6.89e4T^{2} \)
43 \( 1 + 332.T + 7.95e4T^{2} \)
47 \( 1 + (156. + 271. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-52.7 + 91.4i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-232. + 402. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-16.4 - 28.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-27.8 + 48.3i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 19.6T + 3.57e5T^{2} \)
73 \( 1 + (83.5 - 144. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-269. - 466. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 597.T + 5.71e5T^{2} \)
89 \( 1 + (579. + 1.00e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 755.T + 9.12e5T^{2} \)
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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

−15.68912081687268508495627099958, −14.57183671649479958623420437302, −12.76723389564385256410748348062, −11.95568112635381279757685576007, −10.36523575739361282912462443929, −9.949500572485253416732943072941, −8.424135882635389639593586364505, −6.38502109874355050343796155287, −4.92438563170199434697571499939, −3.81688973236217745125067470862, 0.22960014033445751798513831365, 2.78354873403312429436899038300, 5.53898666112228370249734092235, 6.81809555074545914571052033717, 7.51871743504107657085953621398, 9.422539146223789423465330733638, 11.20982229448801632231001918647, 11.77253421346766496727629253951, 13.09353555289823136920684575072, 13.70984667809903523179843932461

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