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Properties

Label 2-56-56.37-c1-0-3
Degree $2$
Conductor $56$
Sign $0.723 - 0.690i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.804 + 1.16i)2-s + (0.591 − 0.341i)3-s + (−0.706 + 1.87i)4-s + (−2.80 − 1.61i)5-s + (0.872 + 0.413i)6-s + (1.47 − 2.19i)7-s + (−2.74 + 0.682i)8-s + (−1.26 + 2.19i)9-s + (−0.371 − 4.56i)10-s + (2.08 − 1.20i)11-s + (0.220 + 1.34i)12-s + 3.09i·13-s + (3.74 − 0.0456i)14-s − 2.21·15-s + (−3.00 − 2.64i)16-s + (−1.97 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.568 + 0.822i)2-s + (0.341 − 0.197i)3-s + (−0.353 + 0.935i)4-s + (−1.25 − 0.724i)5-s + (0.356 + 0.168i)6-s + (0.558 − 0.829i)7-s + (−0.970 + 0.241i)8-s + (−0.422 + 0.731i)9-s + (−0.117 − 1.44i)10-s + (0.629 − 0.363i)11-s + (0.0637 + 0.388i)12-s + 0.858i·13-s + (0.999 − 0.0122i)14-s − 0.570·15-s + (−0.750 − 0.661i)16-s + (−0.479 − 0.830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953947 + 0.381943i\)
\(L(\frac12)\) \(\approx\) \(0.953947 + 0.381943i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.804 - 1.16i)T \)
7 \( 1 + (-1.47 + 2.19i)T \)
good3 \( 1 + (-0.591 + 0.341i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.80 + 1.61i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.09iT - 13T^{2} \)
17 \( 1 + (1.97 + 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.33 - 1.35i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.37 - 2.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (-1.10 - 1.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.30 - 2.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.11T + 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 + (-3.31 + 5.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.23 + 1.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.6 - 6.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.54 + 4.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.01 + 2.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + (-4.77 - 8.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.838 - 1.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.47iT - 83T^{2} \)
89 \( 1 + (6.98 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.37T + 97T^{2} \)
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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

−15.46613373779745623669060178907, −14.11726999185951822939095129095, −13.63636895054855778329494541082, −12.08114503732660925245169197148, −11.31908299054326831082300387313, −8.934166148338144865198683193169, −7.993450539690396583703630937569, −7.08804864379128851419780938401, −4.98371421535595856702833118562, −3.84261663580936754277667566102, 2.96164886831542471105044426293, 4.27675426981475576534549541065, 6.17873705041384779909036676041, 8.103577361624593338096473943352, 9.391931185142906433250929647582, 10.93245395930382222426295018590, 11.73086345513602739966929778280, 12.57540177689535561168661273329, 14.29882268124607359919298376859, 15.07202954742350747115027865598

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