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Properties

Label 1-643-643.34-r0-0-0
Degree $1$
Conductor $643$
Sign $0.642 - 0.766i$
Analytic cond. $2.98607$
Root an. cond. $2.98607$
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  + (−0.246 − 0.969i)2-s + (−0.303 + 0.952i)3-s + (−0.877 + 0.478i)4-s + (−0.465 + 0.884i)5-s + (0.998 + 0.0586i)6-s + (0.708 + 0.705i)7-s + (0.680 + 0.732i)8-s + (−0.815 − 0.578i)9-s + (0.972 + 0.232i)10-s + (−0.628 − 0.777i)11-s + (−0.189 − 0.981i)12-s + (0.0635 − 0.997i)13-s + (0.508 − 0.861i)14-s + (−0.701 − 0.712i)15-s + (0.541 − 0.840i)16-s + (−0.377 − 0.926i)17-s + ⋯
L(s)  = 1  + (−0.246 − 0.969i)2-s + (−0.303 + 0.952i)3-s + (−0.877 + 0.478i)4-s + (−0.465 + 0.884i)5-s + (0.998 + 0.0586i)6-s + (0.708 + 0.705i)7-s + (0.680 + 0.732i)8-s + (−0.815 − 0.578i)9-s + (0.972 + 0.232i)10-s + (−0.628 − 0.777i)11-s + (−0.189 − 0.981i)12-s + (0.0635 − 0.997i)13-s + (0.508 − 0.861i)14-s + (−0.701 − 0.712i)15-s + (0.541 − 0.840i)16-s + (−0.377 − 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(2.98607\)
Root analytic conductor: \(2.98607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (0:\ ),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6969378854 - 0.3249714338i\)
\(L(\frac12)\) \(\approx\) \(0.6969378854 - 0.3249714338i\)
\(L(1)\) \(\approx\) \(0.7175486074 - 0.09320527365i\)
\(L(1)\) \(\approx\) \(0.7175486074 - 0.09320527365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.246 - 0.969i)T \)
3 \( 1 + (-0.303 + 0.952i)T \)
5 \( 1 + (-0.465 + 0.884i)T \)
7 \( 1 + (0.708 + 0.705i)T \)
11 \( 1 + (-0.628 - 0.777i)T \)
13 \( 1 + (0.0635 - 0.997i)T \)
17 \( 1 + (-0.377 - 0.926i)T \)
19 \( 1 + (0.810 - 0.586i)T \)
23 \( 1 + (-0.959 + 0.280i)T \)
29 \( 1 + (0.987 - 0.155i)T \)
31 \( 1 + (-0.265 - 0.964i)T \)
37 \( 1 + (0.605 + 0.795i)T \)
41 \( 1 + (0.873 - 0.487i)T \)
43 \( 1 + (0.863 + 0.504i)T \)
47 \( 1 + (0.256 - 0.966i)T \)
53 \( 1 + (-0.827 + 0.562i)T \)
59 \( 1 + (-0.673 + 0.739i)T \)
61 \( 1 + (0.821 - 0.570i)T \)
67 \( 1 + (0.636 - 0.771i)T \)
71 \( 1 + (0.439 - 0.898i)T \)
73 \( 1 + (0.0830 + 0.996i)T \)
79 \( 1 + (-0.997 - 0.0684i)T \)
83 \( 1 + (-0.265 + 0.964i)T \)
89 \( 1 + (-0.715 - 0.698i)T \)
97 \( 1 + (0.786 - 0.617i)T \)
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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

−23.58412345682031332978214650557, −22.65649532385961326352307347246, −21.41181678047957438574436094085, −20.258187330022922305084104300045, −19.62758753171968951885670345023, −18.706971718986029577145185394659, −17.71235848448161337284032049825, −17.43832274713043346221978866937, −16.35044827484920177244808637045, −15.903936129106714095918880882305, −14.46566818118087671244783440383, −14.02328835606564855252577976772, −12.94048888152965861493360257789, −12.360130235715976616005250827159, −11.24521415355558006785015716211, −10.205416527686903912022839306023, −8.993584998558713739433042436841, −8.069190214240657291122081572505, −7.64662469574719401094762583129, −6.75344134877070337212239285902, −5.70227718266766772904475918217, −4.75748114720120170061197666906, −4.066345672674283821922025492057, −1.87404487628392049511938262396, −1.02759518475086723930895963877, 0.5480631065598425281346729174, 2.5879200244568072296924541067, 2.99622639809100348773187762518, 4.1388157359385949977001576209, 5.1080040046220050278794765599, 5.92735764368034022798312477451, 7.67513350209815868665150071437, 8.338206439401111428666048785044, 9.39197935414436492836487815543, 10.2127219280524137794701212131, 11.12757244104725957581519686847, 11.3982850818645149044264608831, 12.29191797209298290529794905134, 13.67108252314985663373913875318, 14.36866314905753835564278615954, 15.50745586111493629697120558389, 15.89628812317971144998764205453, 17.27968681488985168947839668844, 18.161504997958002880351259775949, 18.41188681507925981334529277233, 19.69892369595477431657698935380, 20.368721567211506625886643815, 21.21142995388955854459569950308, 21.95967512780890915627256690658, 22.39926530403575763893703393751

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