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Properties

Label 1-1020-1020.959-r1-0-0
Degree $1$
Conductor $1020$
Sign $0.528 - 0.848i$
Analytic cond. $109.614$
Root an. cond. $109.614$
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  + (−0.923 − 0.382i)7-s + (0.382 − 0.923i)11-s + i·13-s + (0.707 + 0.707i)19-s + (0.382 − 0.923i)23-s + (−0.923 + 0.382i)29-s + (−0.382 − 0.923i)31-s + (0.382 + 0.923i)37-s + (0.923 + 0.382i)41-s + (−0.707 + 0.707i)43-s − i·47-s + (0.707 + 0.707i)49-s + (−0.707 − 0.707i)53-s + (0.707 − 0.707i)59-s + (0.923 + 0.382i)61-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)7-s + (0.382 − 0.923i)11-s + i·13-s + (0.707 + 0.707i)19-s + (0.382 − 0.923i)23-s + (−0.923 + 0.382i)29-s + (−0.382 − 0.923i)31-s + (0.382 + 0.923i)37-s + (0.923 + 0.382i)41-s + (−0.707 + 0.707i)43-s − i·47-s + (0.707 + 0.707i)49-s + (−0.707 − 0.707i)53-s + (0.707 − 0.707i)59-s + (0.923 + 0.382i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.528 - 0.848i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ 0.528 - 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.366480749 - 0.7585857409i\)
\(L(\frac12)\) \(\approx\) \(1.366480749 - 0.7585857409i\)
\(L(1)\) \(\approx\) \(0.9740663325 - 0.09610540808i\)
\(L(1)\) \(\approx\) \(0.9740663325 - 0.09610540808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + (-0.923 - 0.382i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−21.70188770293804216194655301362, −20.62848855042987150886248414866, −19.88453824514248943758715953646, −19.393443009776162148156749007615, −18.31365816081666774370766566047, −17.69498668413425152578849541833, −16.88874618394090115602978081320, −15.856195347575750891161852205427, −15.378092914152866085151487358449, −14.55699171124229440550850516358, −13.43096708280653057541263256020, −12.82613673030394475214139830531, −12.09512912954929398640649407688, −11.17640053465644440865526607244, −10.14654647797012872924056992915, −9.476809262015037714843231255900, −8.77804964604876757036782686035, −7.484709795593464300654487529926, −6.98278829153344260866327491918, −5.808900979525902029368237670546, −5.17176047935876252001642909816, −3.89852675821152697787506232136, −3.090227842535629780428242633573, −2.091922128862897402565445078538, −0.77651653218170795831099962497, 0.44316873902420524465169138014, 1.553795852601407230296326342273, 2.895440702859872787368736219864, 3.670175565855809940246553370, 4.55911745458272067967626179288, 5.85147077592856617390522797394, 6.45805990562055635858517033205, 7.340867319946875068233014563476, 8.35403441727691623340511048965, 9.320332677378692314317050005426, 9.85206738863245243967704690537, 10.9965748609723181779200366064, 11.59655543106168053801196687461, 12.652857858788678575669576089710, 13.34251331845859199531661707387, 14.17567456616448930018716328707, 14.83139904896399505381912980993, 16.18983452532957527290679626346, 16.398683155416601183682565327473, 17.146546930594806640362235652101, 18.43820480115169426641843351453, 18.90990069742189860150853828968, 19.650779490286416156828606341321, 20.51559252047196274627280229622, 21.244968370309495193128329271430

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