Properties

Label 1-712-712.491-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.917 + 0.397i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (0.479 − 0.877i)3-s + (−0.909 − 0.415i)5-s + (−0.349 − 0.936i)7-s + (−0.540 − 0.841i)9-s + (−0.415 − 0.909i)11-s + (−0.877 − 0.479i)13-s + (−0.800 + 0.599i)15-s + (0.989 − 0.142i)17-s + (0.977 + 0.212i)19-s + (−0.989 − 0.142i)21-s + (0.212 − 0.977i)23-s + (0.654 + 0.755i)25-s + (−0.997 + 0.0713i)27-s + (−0.936 + 0.349i)29-s + (0.212 + 0.977i)31-s + ⋯
L(s)  = 1  + (0.479 − 0.877i)3-s + (−0.909 − 0.415i)5-s + (−0.349 − 0.936i)7-s + (−0.540 − 0.841i)9-s + (−0.415 − 0.909i)11-s + (−0.877 − 0.479i)13-s + (−0.800 + 0.599i)15-s + (0.989 − 0.142i)17-s + (0.977 + 0.212i)19-s + (−0.989 − 0.142i)21-s + (0.212 − 0.977i)23-s + (0.654 + 0.755i)25-s + (−0.997 + 0.0713i)27-s + (−0.936 + 0.349i)29-s + (0.212 + 0.977i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.917 + 0.397i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.917 + 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1618285284 - 0.7797101146i\)
\(L(\frac12)\) \(\approx\) \(-0.1618285284 - 0.7797101146i\)
\(L(1)\) \(\approx\) \(0.6668560103 - 0.5498393362i\)
\(L(1)\) \(\approx\) \(0.6668560103 - 0.5498393362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.479 - 0.877i)T \)
5 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (-0.349 - 0.936i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.877 - 0.479i)T \)
17 \( 1 + (0.989 - 0.142i)T \)
19 \( 1 + (0.977 + 0.212i)T \)
23 \( 1 + (0.212 - 0.977i)T \)
29 \( 1 + (-0.936 + 0.349i)T \)
31 \( 1 + (0.212 + 0.977i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.877 + 0.479i)T \)
43 \( 1 + (0.936 + 0.349i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (-0.479 - 0.877i)T \)
61 \( 1 + (-0.0713 - 0.997i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.909 - 0.415i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.540 + 0.841i)T \)
83 \( 1 + (-0.800 - 0.599i)T \)
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

−22.6475828878532842526706135926, −22.42006848304785680567587193100, −21.423683824831852633394500018799, −20.65053060624066472077426879147, −19.784450775250537679375685765412, −19.10590384489285519673729713157, −18.44298478266626627700321562911, −17.182085392861537051832294637400, −16.30340585627343405279886709580, −15.3832447028124602075416157414, −15.15701235538836562419303370532, −14.299246948424770857220453472705, −13.16244466000112326051664746041, −11.93766867130304087955236032533, −11.63345089629778220801476783762, −10.25502141160142561641400604447, −9.701026803949961039903248808719, −8.84445788743699443354239494258, −7.74370533412313787989553829165, −7.19284829804153285499423381594, −5.62550985712552080170432698204, −4.88988025061824780099278729775, −3.80124194860466620303559434220, −3.008285207822753885989044511551, −2.09934821877222585658844860163, 0.37297275629002913460823265368, 1.30271526619969260677948385103, 3.02993093810817689332212519742, 3.425004063822120039149072940246, 4.7793967500008769336449962649, 5.86151859351142835845932868960, 7.14948592196041420140651965650, 7.59971763792415808152888824515, 8.34625387090376144111076058943, 9.355324660005049499962985012252, 10.45671616693078416550820389623, 11.42783794276475275019671131246, 12.43825957238072646301455524727, 12.82880730571928671336500785656, 13.96396330625429650040401343918, 14.45461948123586083797988560384, 15.64225333028748873873398994914, 16.47753295380070739112752822678, 17.13993327647958926195164446031, 18.30866775565472014989298126576, 19.02106190194322543234053437086, 19.68208815054435766783873981825, 20.33526084674839680774095716696, 20.98815850095211666462022400706, 22.44100381532156885421195712837

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