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Properties

Label 1-712-712.691-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.376 - 0.926i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.909 + 0.415i)3-s + (−0.142 − 0.989i)5-s + (−0.989 + 0.142i)7-s + (0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.909 + 0.415i)13-s + (0.540 + 0.841i)15-s + (−0.841 − 0.540i)17-s + (0.755 + 0.654i)19-s + (0.841 − 0.540i)21-s + (0.755 + 0.654i)23-s + (−0.959 + 0.281i)25-s + (−0.281 + 0.959i)27-s + (0.989 − 0.142i)29-s + (0.755 − 0.654i)31-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)3-s + (−0.142 − 0.989i)5-s + (−0.989 + 0.142i)7-s + (0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.909 + 0.415i)13-s + (0.540 + 0.841i)15-s + (−0.841 − 0.540i)17-s + (0.755 + 0.654i)19-s + (0.841 − 0.540i)21-s + (0.755 + 0.654i)23-s + (−0.959 + 0.281i)25-s + (−0.281 + 0.959i)27-s + (0.989 − 0.142i)29-s + (0.755 − 0.654i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.376 - 0.926i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.376 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1727042238 - 0.2565952081i\)
\(L(\frac12)\) \(\approx\) \(0.1727042238 - 0.2565952081i\)
\(L(1)\) \(\approx\) \(0.6032905197 + 0.02881916392i\)
\(L(1)\) \(\approx\) \(0.6032905197 + 0.02881916392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.909 + 0.415i)T \)
5 \( 1 + (-0.142 - 0.989i)T \)
7 \( 1 + (-0.989 + 0.142i)T \)
11 \( 1 + (-0.142 + 0.989i)T \)
13 \( 1 + (-0.909 + 0.415i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + (0.989 - 0.142i)T \)
31 \( 1 + (0.755 - 0.654i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.909 + 0.415i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.909 - 0.415i)T \)
61 \( 1 + (-0.281 + 0.959i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.73054692769014846021013528750, −21.958231717523716260204248664853, −21.4946149945020675092857869618, −19.75873846035579260781258934166, −19.394202191307781523840700436424, −18.57622787962883130763259109398, −17.76240633423473768552879019176, −17.06654643437296919759788363855, −16.02379812210897331134220012492, −15.54412929881631200923418725522, −14.297823945487158984134675330182, −13.45068492623104709908527077906, −12.67120401852255242390364700160, −11.80772403610817242485178675279, −10.81075903984868770199844550155, −10.45243866537067276071346493595, −9.3310661150490661209100096674, −8.0200619304259742558700946653, −6.99423322723603989959614144695, −6.5398155993611759481833389251, −5.65155573004996507493102252237, −4.53495195933088882393290326835, −3.20579337461960445486555578907, −2.48261529072618844080135477074, −0.761382975216383936156875082, 0.12061673094906279086723873298, 1.30337178523170041698577984845, 2.774137244426058286664244653963, 4.171333578610360450701726404, 4.7850549048719808642950849957, 5.63570381938957728355087679680, 6.70139461485036101922101048540, 7.4686041626191328086304117993, 8.90236719818776933585802752891, 9.71246184693604947926881147386, 10.0919728295107051666773922152, 11.60991219699420066946042661370, 12.0610667502580802954149215771, 12.82791822104316212852994731617, 13.638432204643618593304302581757, 15.16316635305701772980908138235, 15.661532458424668284058539157793, 16.49365639038914575171054777252, 17.0903627147372932698436542448, 17.852799261237627520012589863837, 18.884250426907593096637213079573, 19.87576803982974445282438154742, 20.49443562082435187308896091787, 21.451350459128854144873638617992, 22.1996103642768299727809090621

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