Properties

Label 1-712-712.107-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.555 + 0.831i$
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  + (0.281 + 0.959i)3-s + (0.415 − 0.909i)5-s + (0.909 + 0.415i)7-s + (−0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (0.281 + 0.959i)13-s + (0.989 + 0.142i)15-s + (0.142 + 0.989i)17-s + (0.540 + 0.841i)19-s + (−0.142 + 0.989i)21-s + (0.540 + 0.841i)23-s + (−0.654 − 0.755i)25-s + (−0.755 − 0.654i)27-s + (−0.909 − 0.415i)29-s + (0.540 − 0.841i)31-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)3-s + (0.415 − 0.909i)5-s + (0.909 + 0.415i)7-s + (−0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (0.281 + 0.959i)13-s + (0.989 + 0.142i)15-s + (0.142 + 0.989i)17-s + (0.540 + 0.841i)19-s + (−0.142 + 0.989i)21-s + (0.540 + 0.841i)23-s + (−0.654 − 0.755i)25-s + (−0.755 − 0.654i)27-s + (−0.909 − 0.415i)29-s + (0.540 − 0.841i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260726551 + 2.358060977i\)
\(L(\frac12)\) \(\approx\) \(1.260726551 + 2.358060977i\)
\(L(1)\) \(\approx\) \(1.265982883 + 0.6280850419i\)
\(L(1)\) \(\approx\) \(1.265982883 + 0.6280850419i\)

Euler product

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

−22.419237823909973014342671475217, −21.14768178742565104403169292362, −20.49226656196365853132642195486, −19.58353476217835388093003266304, −18.737259579670471547785058748525, −18.040014613244697929228673556447, −17.58043662450858604548818106913, −16.58035486620482549636727008519, −15.26714884197020291342754176031, −14.485315677402665931893542030904, −13.81748931820545492569289263316, −13.32155267343384090795661862591, −12.06582631423275403212627530387, −11.19375967644583489521770464549, −10.67948926398335857568318349246, −9.332336094483508949352725750360, −8.424237015150396731249538681533, −7.53513019738060884523755375164, −6.860561550241101268798465163215, −5.9378343963894840733437232369, −4.96329602622179402826661785195, −3.312867214700872785625280103646, −2.77926702610424671459838953956, −1.48582313919982346237297300813, −0.58460167758814445762559753963, 1.46165324149148665490474594764, 2.12427457074672255241956618945, 3.768674460032525281692043894905, 4.44651099886210575903657822593, 5.28865590069945339958279057169, 6.08742866641683495074494438860, 7.69231739616448655930971718981, 8.43743067810675121992602766745, 9.3802401942475110888492613950, 9.73997089535488272876281848798, 11.03863485259330728705933545800, 11.7552047730491408250041421508, 12.688328411076052582537947905771, 13.79626614245752390258300101518, 14.5370800653559546057368152227, 15.23451737777971720904857918046, 16.12780984476998170481710094303, 17.05800609640887151802208398060, 17.407920217057151845440779280050, 18.640934235975451317959977239969, 19.66621451485986908957958402534, 20.49275471180135910830119420541, 21.112592657133584064045469607517, 21.525354540109396356517338757786, 22.52761245011874607063426774417

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