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Properties

Label 1-712-712.99-r1-0-0
Degree $1$
Conductor $712$
Sign $0.746 - 0.665i$
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  + (0.989 − 0.142i)3-s + (0.841 − 0.540i)5-s + (−0.540 − 0.841i)7-s + (0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (0.989 − 0.142i)13-s + (0.755 − 0.654i)15-s + (0.654 − 0.755i)17-s + (0.281 + 0.959i)19-s + (−0.654 − 0.755i)21-s + (0.281 + 0.959i)23-s + (0.415 − 0.909i)25-s + (0.909 − 0.415i)27-s + (0.540 + 0.841i)29-s + (0.281 − 0.959i)31-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (0.841 − 0.540i)5-s + (−0.540 − 0.841i)7-s + (0.959 − 0.281i)9-s + (0.841 + 0.540i)11-s + (0.989 − 0.142i)13-s + (0.755 − 0.654i)15-s + (0.654 − 0.755i)17-s + (0.281 + 0.959i)19-s + (−0.654 − 0.755i)21-s + (0.281 + 0.959i)23-s + (0.415 − 0.909i)25-s + (0.909 − 0.415i)27-s + (0.540 + 0.841i)29-s + (0.281 − 0.959i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.746 - 0.665i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.746 - 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.099746811 - 1.561480703i\)
\(L(\frac12)\) \(\approx\) \(4.099746811 - 1.561480703i\)
\(L(1)\) \(\approx\) \(1.933885126 - 0.4077518335i\)
\(L(1)\) \(\approx\) \(1.933885126 - 0.4077518335i\)

Euler product

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

−22.16835345658425106513416516963, −21.62437106330713100816241052956, −21.068673537249175105107557591757, −19.99981334784624339566411711344, −19.09283260887871563506655817153, −18.6989029846383342554140756413, −17.76413422825708905058697331082, −16.66773939241884211706795345059, −15.78161953437535197922109515438, −14.9914620215709178432921841143, −14.22555515146360052827519582513, −13.54671496735759874496171906253, −12.78610347544140106003747237422, −11.671291265113285093173997683831, −10.52752422786603837519376248565, −9.82184860229476233698377754393, −8.80102922687345246643986698000, −8.55605824963943889732633549333, −6.95685068318670843997968723982, −6.342764905403670490707640159007, −5.35884942360385661710623046005, −3.90039085400157494444279791397, −3.10278421178841126501371771999, −2.28236184308973087605143622698, −1.18109948991025925299894152034, 1.07582947278869054442822963385, 1.59386544020531254144649848295, 3.07025880436637408917106813802, 3.796141280009590243976764171609, 4.874119063232373290089858468245, 6.16052773791666505223090452825, 6.95797892899709102514318817841, 7.9445188111083700073645878938, 8.8577421667806043869281478896, 9.83695418210207739156009380551, 9.99719123806727663864728554348, 11.56389897331708638765657511354, 12.65718941970108711778523938639, 13.33445449428618219268356791469, 13.964722161474760441455022421593, 14.62170523911667420157238524550, 15.83832082499763235641222259657, 16.52929992682791590244237024728, 17.40056600209444089320111732964, 18.31051047103168859624116366674, 19.14007454340056832887803525326, 20.13002976460361457406117559899, 20.53126019029082514431645614867, 21.19645417882675743685404551269, 22.28118172530347520888900244446

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