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Properties

Label 1-812-812.347-r1-0-0
Degree $1$
Conductor $812$
Sign $0.0633 + 0.997i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (−0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5614307232 + 0.5269315854i\)
\(L(\frac12)\) \(\approx\) \(0.5614307232 + 0.5269315854i\)
\(L(1)\) \(\approx\) \(0.7622891505 - 0.04837501029i\)
\(L(1)\) \(\approx\) \(0.7622891505 - 0.04837501029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−21.7267848234480448493959290193, −20.92917918928380709349241018532, −20.46385658263112819996434619679, −19.68546646122173039980168173429, −18.50780150007973560439502629491, −17.65285249572490961715142807781, −16.92307197783916533730766831446, −16.053769579142177118419468870518, −15.595280353514834723129808807351, −14.857804729738535901168565535761, −13.57934277252861401849496782550, −12.76978498163492718902711956187, −11.85962970111694143983973585703, −11.21935508024867871057625449605, −10.28748889191084313145769100451, −9.35012802136372246049413529517, −8.74202434043651251302462235256, −7.66032280070924937583572514383, −6.618144552882920705372503587326, −5.35836300048889519827661101812, −4.88831328511795127593378235429, −3.96939109969750602074579080470, −3.01292502119438915290991084737, −1.369325513078973092591180748131, −0.23672897035108542670976918796, 0.901741145461681564109977165088, 2.134796386401284095072344529669, 3.189873451847520553731392764676, 4.131591258507628024807962832607, 5.685721218087872046500027122864, 6.14501502186902630980790664087, 7.072119620461204177541936033200, 8.05567457432857238888659380189, 8.48710927969818409423688878459, 10.16444443363676246717531377312, 10.97165235507909395918590580471, 11.34265339761659150088475716043, 12.51811921834330966918660352797, 13.10198155884214055925903880554, 14.135087144184321118827765036335, 14.75974700068244382151211647568, 15.93546504068247136034561643774, 16.57052068018117053904820681492, 17.52200771247713770098369882006, 18.55938183299981002899191093007, 18.7564893664518066416978526123, 19.46858579700034480521072858080, 20.645110761429366718731919224653, 21.52203450487988214860795006572, 22.44318403196456501640993915798

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