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Properties

Label 1-712-712.349-r1-0-0
Degree $1$
Conductor $712$
Sign $0.732 - 0.680i$
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  + (0.877 − 0.479i)3-s + (0.909 − 0.415i)5-s + (0.936 + 0.349i)7-s + (0.540 − 0.841i)9-s + (0.415 − 0.909i)11-s + (0.479 + 0.877i)13-s + (0.599 − 0.800i)15-s + (−0.989 − 0.142i)17-s + (0.212 + 0.977i)19-s + (0.989 − 0.142i)21-s + (0.977 − 0.212i)23-s + (0.654 − 0.755i)25-s + (0.0713 − 0.997i)27-s + (−0.349 + 0.936i)29-s + (0.977 + 0.212i)31-s + ⋯
L(s)  = 1  + (0.877 − 0.479i)3-s + (0.909 − 0.415i)5-s + (0.936 + 0.349i)7-s + (0.540 − 0.841i)9-s + (0.415 − 0.909i)11-s + (0.479 + 0.877i)13-s + (0.599 − 0.800i)15-s + (−0.989 − 0.142i)17-s + (0.212 + 0.977i)19-s + (0.989 − 0.142i)21-s + (0.977 − 0.212i)23-s + (0.654 − 0.755i)25-s + (0.0713 − 0.997i)27-s + (−0.349 + 0.936i)29-s + (0.977 + 0.212i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.732 - 0.680i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.732 - 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.278612955 - 1.680284902i\)
\(L(\frac12)\) \(\approx\) \(4.278612955 - 1.680284902i\)
\(L(1)\) \(\approx\) \(1.998209053 - 0.4586343478i\)
\(L(1)\) \(\approx\) \(1.998209053 - 0.4586343478i\)

Euler product

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

−22.40879973910255649974631522900, −21.443033705160491016308533469520, −20.94369337208768444641096896007, −20.150810763940871830100922896386, −19.498678736525558060982532790255, −18.22762987665488831506789771542, −17.66493886642554798643075588337, −16.97801391860934269047771446065, −15.57558466101591040701015148971, −15.03186789029715828369434658803, −14.35793894408145187909613191046, −13.422269739140642022753222608985, −12.99307210544005460594622125033, −11.29857586666881273038469891880, −10.76493538702753515501645513894, −9.74836539274938306457342032345, −9.16784777467957780818332766858, −8.10854683816979037714161325001, −7.30112035946401552361100568039, −6.26290259477921857029138443354, −4.94631393665199253343359889818, −4.35790737869001481484772081532, −3.02852871881856309416069462078, −2.20934691126482264607258418487, −1.21795347299119420838645442731, 1.08904221495855085973518932549, 1.74299830353344859855010963725, 2.715422068714602673237813967905, 3.93784993829906814388777290674, 4.99932787618545882696605038017, 6.10797191910117908454418130408, 6.86738715199037232017174402931, 8.14355245229927503008407326147, 8.832095712708018312179250850243, 9.25351627398674008705705398061, 10.56430492081863342390828446053, 11.55565607624055010656048410234, 12.4429696629844287874897248971, 13.47272043413754253408787130782, 13.96672338262605264236005113594, 14.605466238517473733657839830764, 15.63395652709027579588897574251, 16.71326794852163710307669755048, 17.45972340732118998423169003391, 18.52670507567479855201214345288, 18.75141701695456594144761577196, 20.04717105570173125218624237895, 20.68612295959132715291136162420, 21.39299435076194601060113853419, 21.929290673850933200061365585818

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