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Properties

Label 1-712-712.307-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.106 + 0.994i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.106 + 0.994i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.106 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.047409103 + 1.165207326i\)
\(L(\frac12)\) \(\approx\) \(1.047409103 + 1.165207326i\)
\(L(1)\) \(\approx\) \(0.8257962960 + 0.3977632288i\)
\(L(1)\) \(\approx\) \(0.8257962960 + 0.3977632288i\)

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

−22.65676029301083063952739358887, −21.10941219403762901144289323177, −20.50664480243965023193373461616, −19.781004755901931281367780833184, −18.90780312372895809354277652668, −17.97408467445961946112310481249, −17.34606124641344338620070677841, −16.63794441143208012476819575038, −15.74013789108085727251827569911, −14.72279184248139742015635437962, −13.78789569618230840535907429737, −12.7968533282480945020400134186, −12.35015628138594182654435532134, −11.44721450394926901471780600222, −10.65932795150712532266469701440, −9.60857876247307500232905291701, −8.25877768282933994219113361249, −7.47352019832562032801231275051, −7.32688023992539284277065648837, −5.41397049006672137568822692194, −5.2201372890305910760669457835, −4.00429603682973973774881221645, −2.61236141724406311733053632549, −1.2712263209870609302552023025, −0.62622293374494478481677040799, 0.775028343405058935771803851501, 2.552647921260022937275445093370, 3.36924050731562520230421083928, 4.496910788685308150813696267299, 5.281853128014346761883415355007, 6.18156929607437433332611999003, 7.31688272751265373645188918605, 8.24367180510940340233255703827, 9.198089062212415325442449072310, 10.20337200948937396153223751232, 11.020236721027714330536611432596, 11.651208977982808626472319911600, 12.225012994625476941631538689061, 13.77551510558508126352993487807, 14.650375343216974608974372195127, 15.221531960386410336408094908396, 15.98712971232630607972018982521, 16.79798124779252270032294430927, 17.69791836108752127466319557718, 18.68558528628286222484871744953, 19.06565624375940146995485201800, 20.41908628566709016898719470658, 21.194878451561825601462512493352, 21.78837009175534682536715056953, 22.479100533689091442570504342464

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