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Properties

Label 1-712-712.227-r1-0-0
Degree $1$
Conductor $712$
Sign $0.155 + 0.987i$
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  + (0.540 − 0.841i)3-s + (−0.654 + 0.755i)5-s + (−0.755 − 0.654i)7-s + (−0.415 − 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.540 − 0.841i)13-s + (0.281 + 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (−0.959 + 0.281i)21-s + (−0.909 + 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (0.755 + 0.654i)29-s + (−0.909 − 0.415i)31-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)3-s + (−0.654 + 0.755i)5-s + (−0.755 − 0.654i)7-s + (−0.415 − 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.540 − 0.841i)13-s + (0.281 + 0.959i)15-s + (0.959 + 0.281i)17-s + (−0.909 + 0.415i)19-s + (−0.959 + 0.281i)21-s + (−0.909 + 0.415i)23-s + (−0.142 − 0.989i)25-s + (−0.989 − 0.142i)27-s + (0.755 + 0.654i)29-s + (−0.909 − 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.155 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2369436397 + 0.2025462503i\)
\(L(\frac12)\) \(\approx\) \(0.2369436397 + 0.2025462503i\)
\(L(1)\) \(\approx\) \(0.8000042473 - 0.2635459486i\)
\(L(1)\) \(\approx\) \(0.8000042473 - 0.2635459486i\)

Euler product

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

−22.032090471226925142268300400415, −21.32491761160856541839373400814, −20.5930047211980604744055348137, −19.90211394511382007233588326076, −19.1014949967585585199816894311, −18.3908636134181853252430297220, −16.93325183469925260095324417358, −16.33154182568063366209236670047, −15.62977669090190750467920380777, −15.10955887440562950009009496809, −14.01625354981155688643890104768, −13.05474793864072968495578343352, −12.26543316129918493050101405863, −11.42927174251713449972360284614, −10.2123861695964786811310140482, −9.58848986114853121536974019976, −8.64962004758836499215513752531, −8.13438636098916862618462212676, −6.895393337783384179002417669125, −5.63693113530828175855775548226, −4.717651655751875772565500425855, −3.962889855900719753227287454, −2.96212732113087381516225990963, −1.88687108295793544208623690477, −0.07966722340038188773094430705, 0.88654961803930610725726549158, 2.37070200875081035188445586657, 3.43400078994124849550452060115, 3.74094458016680562450504486637, 5.71161201053306428097724336985, 6.40079695988683131711032859293, 7.45117412066763139561237586866, 7.93405466593025841884783028027, 8.83664993807796428387064249059, 10.26372487256444377848854698395, 10.69736856021054540596592723794, 11.93957857932425347495633190944, 12.716177656284267433690301315829, 13.50342390358914632351691032417, 14.23608547592552397831464573847, 15.09759637941273045968775876260, 15.96029716874215357020256181027, 16.82059615546953588050120835362, 18.061463176991906305659291298823, 18.55649656554509505944202773365, 19.385466884326475413954589680752, 19.86286883578354327395763036638, 20.78256866191515216569400127373, 21.805284351796497229429193141845, 22.89495174642210920763123276041

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