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Properties

Label 2-4140-15.8-c1-0-25
Degree $2$
Conductor $4140$
Sign $0.0769 + 0.997i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.994 − 2.00i)5-s + (−2.91 + 2.91i)7-s + 4.72i·11-s + (0.839 + 0.839i)13-s + (−5.22 − 5.22i)17-s + 3.12i·19-s + (−0.707 + 0.707i)23-s + (−3.02 + 3.98i)25-s + 4.03·29-s − 6.68·31-s + (8.74 + 2.94i)35-s + (2.81 − 2.81i)37-s − 9.02i·41-s + (1.64 + 1.64i)43-s + (−1.14 − 1.14i)47-s + ⋯
L(s)  = 1  + (−0.444 − 0.895i)5-s + (−1.10 + 1.10i)7-s + 1.42i·11-s + (0.232 + 0.232i)13-s + (−1.26 − 1.26i)17-s + 0.716i·19-s + (−0.147 + 0.147i)23-s + (−0.604 + 0.796i)25-s + 0.748·29-s − 1.20·31-s + (1.47 + 0.497i)35-s + (0.463 − 0.463i)37-s − 1.40i·41-s + (0.251 + 0.251i)43-s + (−0.167 − 0.167i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0769 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0769 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.0769 + 0.997i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.0769 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6296900743\)
\(L(\frac12)\) \(\approx\) \(0.6296900743\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.994 + 2.00i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (2.91 - 2.91i)T - 7iT^{2} \)
11 \( 1 - 4.72iT - 11T^{2} \)
13 \( 1 + (-0.839 - 0.839i)T + 13iT^{2} \)
17 \( 1 + (5.22 + 5.22i)T + 17iT^{2} \)
19 \( 1 - 3.12iT - 19T^{2} \)
29 \( 1 - 4.03T + 29T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + (-2.81 + 2.81i)T - 37iT^{2} \)
41 \( 1 + 9.02iT - 41T^{2} \)
43 \( 1 + (-1.64 - 1.64i)T + 43iT^{2} \)
47 \( 1 + (1.14 + 1.14i)T + 47iT^{2} \)
53 \( 1 + (0.243 - 0.243i)T - 53iT^{2} \)
59 \( 1 - 3.07T + 59T^{2} \)
61 \( 1 - 8.53T + 61T^{2} \)
67 \( 1 + (4.44 - 4.44i)T - 67iT^{2} \)
71 \( 1 - 5.08iT - 71T^{2} \)
73 \( 1 + (1.30 + 1.30i)T + 73iT^{2} \)
79 \( 1 - 0.837iT - 79T^{2} \)
83 \( 1 + (-1.87 + 1.87i)T - 83iT^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (-5.02 + 5.02i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.448078950691197557642334489943, −7.37282789532780752202721539242, −6.91499993047796566670937446415, −5.95245915861572584228787216170, −5.25247396756312490670213326140, −4.47968849632027059964883869486, −3.73763926838883892707701129270, −2.59627803924064635691857586673, −1.84560905106697122670095457125, −0.23425971178448327334062325136, 0.818582787137592775957807909872, 2.43730709195529049517507794189, 3.38873325334410563020898499571, 3.73242392297896831820920949672, 4.65082350430080305798007576796, 6.05070019224428115635704275295, 6.39175673698942651475187606870, 6.98487410463120169109817158082, 7.85371525099997346143488652287, 8.496782386793954393695524863180

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