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Properties

Label 1-712-712.333-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: $\chi_{712} (333, \cdot )$
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

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

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