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Properties

Label 2-56-8.5-c1-0-1
Degree $2$
Conductor $56$
Sign $-i$
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  + 1.41i·2-s + 1.41i·3-s − 2.00·4-s − 1.41i·5-s − 2.00·6-s + 7-s − 2.82i·8-s + 0.999·9-s + 2.00·10-s + 2.82i·11-s − 2.82i·12-s − 4.24i·13-s + 1.41i·14-s + 2.00·15-s + 4.00·16-s − 6·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + 0.816i·3-s − 1.00·4-s − 0.632i·5-s − 0.816·6-s + 0.377·7-s − 1.00i·8-s + 0.333·9-s + 0.632·10-s + 0.852i·11-s − 0.816i·12-s − 1.17i·13-s + 0.377i·14-s + 0.516·15-s + 1.00·16-s − 1.45·17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578009 + 0.578009i\)
\(L(\frac12)\) \(\approx\) \(0.578009 + 0.578009i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
7 \( 1 - T \)
good3 \( 1 - 1.41iT - 3T^{2} \)
5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 - 12.7iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−15.49930421278808901056139214054, −14.96452140700649237510354520469, −13.44118598925179590146497719669, −12.53347172894226953103425392872, −10.62691907987965616656245148422, −9.492639763691787647890359277125, −8.442592762612388001199889949344, −7.02844486450375570678104241254, −5.20312920827363574835035017621, −4.29346346328765948086659099522, 2.05976071722565345195527791409, 4.13452873933357371408227588941, 6.28897480723706332234728122835, 7.84323095250001045709995789706, 9.210651331591415488337301882557, 10.69373036516441500804947735799, 11.55515775923201473995815722510, 12.66468291984322253013169299650, 13.76257516078855623119689374270, 14.45766830375256182878283408496

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