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Properties

Label 1-712-712.619-r1-0-0
Degree $1$
Conductor $712$
Sign $0.813 + 0.582i$
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  + (0.654 − 0.755i)3-s + (0.959 − 0.281i)5-s + (−0.959 + 0.281i)7-s + (−0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.654 + 0.755i)13-s + (0.415 − 0.909i)15-s + (0.415 + 0.909i)17-s + (0.142 + 0.989i)19-s + (−0.415 + 0.909i)21-s + (−0.142 − 0.989i)23-s + (0.841 − 0.540i)25-s + (−0.841 − 0.540i)27-s + (−0.959 + 0.281i)29-s + (−0.142 + 0.989i)31-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (0.959 − 0.281i)5-s + (−0.959 + 0.281i)7-s + (−0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.654 + 0.755i)13-s + (0.415 − 0.909i)15-s + (0.415 + 0.909i)17-s + (0.142 + 0.989i)19-s + (−0.415 + 0.909i)21-s + (−0.142 − 0.989i)23-s + (0.841 − 0.540i)25-s + (−0.841 − 0.540i)27-s + (−0.959 + 0.281i)29-s + (−0.142 + 0.989i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.813 + 0.582i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.813 + 0.582i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.918436540 + 0.6158619103i\)
\(L(\frac12)\) \(\approx\) \(1.918436540 + 0.6158619103i\)
\(L(1)\) \(\approx\) \(1.245179206 - 0.1547971408i\)
\(L(1)\) \(\approx\) \(1.245179206 - 0.1547971408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.654 - 0.755i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.654 + 0.755i)T \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.959 + 0.281i)T \)
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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

−22.29696039978594835532455973388, −21.50686316056524111382038653330, −20.64919476573940286925311624014, −20.08725784351196706473924127128, −19.15701795500235886023518326018, −18.286863727927576072005074192727, −17.328462381658378414999003060255, −16.54546455792486104722712763637, −15.587390930594007518885648062712, −15.0794045729060990031555392823, −13.92848423988498228697083930936, −13.39270236960029839393748783630, −12.682791699478457613502406827953, −11.151883975839979448354511282957, −10.3404617917563249298925930953, −9.578737672573352769213696777229, −9.30195720750759157192199712082, −7.74992183523701822297262983573, −7.16807039158552742454895390259, −5.70387744369413122723300901060, −5.16832352938821732693049234618, −3.86443522439428011848072297450, −2.77332201724017122437230811994, −2.36773238920659034235330734656, −0.434066798691752445841768705619, 1.05721772914095747140393498574, 2.20340104220960994015416688342, 2.833180075397293774512298931677, 4.04980423250959418361237632211, 5.56409590967898885948616209980, 6.16133818801927331942250004372, 7.10362825985534561035436743247, 8.1299972346888945074607408754, 8.98831808983453942445716748973, 9.74535633933830006232013696513, 10.50885427842029971023405108737, 12.029044005715179384878298706621, 12.84374809804315264908661618193, 13.10761257236817755899591708338, 14.29986155378267102557442257680, 14.696917716198880263233660281001, 16.11307737460103642920907289287, 16.69129585491664031572808939923, 17.76054629114922108798155509191, 18.58000765322911285204915644228, 19.08006774293258120225226779851, 19.99024535649087541969646294083, 20.89904948210543344757737926461, 21.492797764627623789816018006444, 22.4362994338373443898581458578

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