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Properties

Label 1-712-712.325-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.656 + 0.754i$
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  + (−0.599 + 0.800i)3-s + (0.540 + 0.841i)5-s + (0.977 + 0.212i)7-s + (−0.281 − 0.959i)9-s + (0.841 + 0.540i)11-s + (0.800 + 0.599i)13-s + (−0.997 − 0.0713i)15-s + (0.755 + 0.654i)17-s + (−0.479 + 0.877i)19-s + (−0.755 + 0.654i)21-s + (−0.877 − 0.479i)23-s + (−0.415 + 0.909i)25-s + (0.936 + 0.349i)27-s + (0.212 − 0.977i)29-s + (−0.877 + 0.479i)31-s + ⋯
L(s)  = 1  + (−0.599 + 0.800i)3-s + (0.540 + 0.841i)5-s + (0.977 + 0.212i)7-s + (−0.281 − 0.959i)9-s + (0.841 + 0.540i)11-s + (0.800 + 0.599i)13-s + (−0.997 − 0.0713i)15-s + (0.755 + 0.654i)17-s + (−0.479 + 0.877i)19-s + (−0.755 + 0.654i)21-s + (−0.877 − 0.479i)23-s + (−0.415 + 0.909i)25-s + (0.936 + 0.349i)27-s + (0.212 − 0.977i)29-s + (−0.877 + 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.656 + 0.754i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003628604 + 2.202761874i\)
\(L(\frac12)\) \(\approx\) \(1.003628604 + 2.202761874i\)
\(L(1)\) \(\approx\) \(1.046361628 + 0.6633865675i\)
\(L(1)\) \(\approx\) \(1.046361628 + 0.6633865675i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.599 + 0.800i)T \)
5 \( 1 + (0.540 + 0.841i)T \)
7 \( 1 + (0.977 + 0.212i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (0.800 + 0.599i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.479 + 0.877i)T \)
23 \( 1 + (-0.877 - 0.479i)T \)
29 \( 1 + (0.212 - 0.977i)T \)
31 \( 1 + (-0.877 + 0.479i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.800 - 0.599i)T \)
43 \( 1 + (0.212 + 0.977i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (0.989 - 0.142i)T \)
59 \( 1 + (0.599 + 0.800i)T \)
61 \( 1 + (-0.349 + 0.936i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (0.540 - 0.841i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (-0.997 + 0.0713i)T \)
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

−21.98623159901804698051565223876, −21.4382565486756704853737671828, −20.31057482905103232702499956798, −19.85657855748909593070449452853, −18.54663084471007132121834089342, −18.02848546092626472947348663082, −17.17236396509452688573112884804, −16.703853122514812101410697654865, −15.7413977112104302958027323463, −14.34663438833531498351795909512, −13.76449170336382530369618278375, −12.99672565055535205972601667656, −12.0969019930675738141971591833, −11.35799866311384914315339918901, −10.610525138018121572500348903228, −9.31720918427188856957772830495, −8.423803986040684327401353347065, −7.71951730192593903464638726241, −6.58340891301073866627333263069, −5.65948434139382681101883581751, −5.07392496825042202584182353985, −3.89845063373874076339435387183, −2.30527191345758460882443468907, −1.24822346459988524870092457390, −0.70282737505873266686912751893, 1.27512965944171544803215073381, 2.27082909178188713593933993062, 3.82377291789089808832445625534, 4.27734101201249392642908855137, 5.76467893894936416597309240581, 6.06513540530382124518925926396, 7.25026568904964222363268163301, 8.45342926468625158908530904852, 9.39794298701661850559068821408, 10.25861798969402423288570868122, 10.92105887878944317012865483097, 11.70718754953735060223159755965, 12.47458373371816128061558486086, 14.00892272583818610399188586502, 14.564022605181573003595965593274, 15.09159665112031737460365453099, 16.27491454721357621508004948797, 16.99939581185564224070724359244, 17.82173983172872585397929409665, 18.33252972332220593398952243156, 19.38492941888882654822980111471, 20.619729891821599064230402063611, 21.2563100363579591514717082814, 21.73132804640967417334165511858, 22.69996439912632466749157863847

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