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Properties

Label 2-4140-15.2-c1-0-30
Degree $2$
Conductor $4140$
Sign $-0.861 + 0.507i$
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  + (−2.21 + 0.331i)5-s + (−1.17 − 1.17i)7-s + 0.236i·11-s + (0.337 − 0.337i)13-s + (0.583 − 0.583i)17-s + 2.36i·19-s + (0.707 + 0.707i)23-s + (4.78 − 1.46i)25-s + 1.54·29-s + 2.36·31-s + (2.97 + 2.20i)35-s + (4.63 + 4.63i)37-s + 3.08i·41-s + (5.19 − 5.19i)43-s + (−3.52 + 3.52i)47-s + ⋯
L(s)  = 1  + (−0.988 + 0.148i)5-s + (−0.442 − 0.442i)7-s + 0.0714i·11-s + (0.0935 − 0.0935i)13-s + (0.141 − 0.141i)17-s + 0.541i·19-s + (0.147 + 0.147i)23-s + (0.956 − 0.292i)25-s + 0.286·29-s + 0.423·31-s + (0.503 + 0.372i)35-s + (0.761 + 0.761i)37-s + 0.482i·41-s + (0.791 − 0.791i)43-s + (−0.514 + 0.514i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.861 + 0.507i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3693855226\)
\(L(\frac12)\) \(\approx\) \(0.3693855226\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.331i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1.17 + 1.17i)T + 7iT^{2} \)
11 \( 1 - 0.236iT - 11T^{2} \)
13 \( 1 + (-0.337 + 0.337i)T - 13iT^{2} \)
17 \( 1 + (-0.583 + 0.583i)T - 17iT^{2} \)
19 \( 1 - 2.36iT - 19T^{2} \)
29 \( 1 - 1.54T + 29T^{2} \)
31 \( 1 - 2.36T + 31T^{2} \)
37 \( 1 + (-4.63 - 4.63i)T + 37iT^{2} \)
41 \( 1 - 3.08iT - 41T^{2} \)
43 \( 1 + (-5.19 + 5.19i)T - 43iT^{2} \)
47 \( 1 + (3.52 - 3.52i)T - 47iT^{2} \)
53 \( 1 + (8.94 + 8.94i)T + 53iT^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 1.02T + 61T^{2} \)
67 \( 1 + (9.64 + 9.64i)T + 67iT^{2} \)
71 \( 1 - 4.90iT - 71T^{2} \)
73 \( 1 + (4.17 - 4.17i)T - 73iT^{2} \)
79 \( 1 - 1.47iT - 79T^{2} \)
83 \( 1 + (-6.21 - 6.21i)T + 83iT^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (5.69 + 5.69i)T + 97iT^{2} \)
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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

−8.030023083054960875449030498949, −7.46228756768845463780029235545, −6.70757466434120116717590042531, −6.05639007214321204618076133721, −4.95923694647908195352476107291, −4.26408117532361579234639707733, −3.46514823829998314025280568334, −2.80039013770668096392514692664, −1.34737019561423915563443224702, −0.12143350172229814788809961200, 1.15188131766961903833891265343, 2.59640774938691324771512305256, 3.26735776662775499460575194169, 4.24695460173890579601469776466, 4.81764257741671083094596316802, 5.88818076047964983828196416930, 6.47692069836510482803462440204, 7.44936355103536989993391727949, 7.86340992496378024879732108681, 8.854992687568187047344574515974

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