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Properties

Label 2-4140-15.8-c1-0-28
Degree $2$
Conductor $4140$
Sign $0.852 + 0.523i$
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  + (2.20 + 0.371i)5-s + (−3.41 + 3.41i)7-s − 4.26i·11-s + (−1.36 − 1.36i)13-s + (−3.76 − 3.76i)17-s + 3.34i·19-s + (0.707 − 0.707i)23-s + (4.72 + 1.63i)25-s + 4.28·29-s + 1.17·31-s + (−8.80 + 6.26i)35-s + (−1.12 + 1.12i)37-s − 10.1i·41-s + (8.84 + 8.84i)43-s + (4.85 + 4.85i)47-s + ⋯
L(s)  = 1  + (0.986 + 0.166i)5-s + (−1.29 + 1.29i)7-s − 1.28i·11-s + (−0.379 − 0.379i)13-s + (−0.912 − 0.912i)17-s + 0.768i·19-s + (0.147 − 0.147i)23-s + (0.944 + 0.327i)25-s + 0.796·29-s + 0.211·31-s + (−1.48 + 1.05i)35-s + (−0.184 + 0.184i)37-s − 1.58i·41-s + (1.34 + 1.34i)43-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.852 + 0.523i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.852 + 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.638802945\)
\(L(\frac12)\) \(\approx\) \(1.638802945\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 0.371i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (3.41 - 3.41i)T - 7iT^{2} \)
11 \( 1 + 4.26iT - 11T^{2} \)
13 \( 1 + (1.36 + 1.36i)T + 13iT^{2} \)
17 \( 1 + (3.76 + 3.76i)T + 17iT^{2} \)
19 \( 1 - 3.34iT - 19T^{2} \)
29 \( 1 - 4.28T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (1.12 - 1.12i)T - 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (-8.84 - 8.84i)T + 43iT^{2} \)
47 \( 1 + (-4.85 - 4.85i)T + 47iT^{2} \)
53 \( 1 + (-7.95 + 7.95i)T - 53iT^{2} \)
59 \( 1 + 4.65T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-9.27 + 9.27i)T - 67iT^{2} \)
71 \( 1 + 7.18iT - 71T^{2} \)
73 \( 1 + (0.225 + 0.225i)T + 73iT^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 + (-7.70 + 7.70i)T - 83iT^{2} \)
89 \( 1 + 3.97T + 89T^{2} \)
97 \( 1 + (0.874 - 0.874i)T - 97iT^{2} \)
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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

−8.668393237843769307416595638317, −7.60397376295252292159355637149, −6.56418698032583333479368351962, −6.13784891236097387140161556647, −5.62108036162719842145565308906, −4.80644802074994496330611709760, −3.40394927522483021162425559922, −2.81822832010986919223467181077, −2.16327732186545202300093389224, −0.55487220752481726551942744169, 0.919424821643384597585398497951, 2.09898476061844134848530797215, 2.89275213713910311514836651956, 4.15830418112882713042891363399, 4.48432450112041824763631090331, 5.60722661986006912141730653700, 6.52637535883717925099296912620, 6.87794326997076443663327375266, 7.47602881241954136598392165493, 8.689816709072799436027117336680

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