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Properties

Label 2-4140-15.8-c1-0-17
Degree $2$
Conductor $4140$
Sign $-0.518 - 0.854i$
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  + (1.79 + 1.33i)5-s + (−2.07 + 2.07i)7-s + 3.22i·11-s + (4.92 + 4.92i)13-s + (3.44 + 3.44i)17-s − 1.09i·19-s + (−0.707 + 0.707i)23-s + (1.41 + 4.79i)25-s + 6.42·29-s − 5.08·31-s + (−6.49 + 0.941i)35-s + (3.18 − 3.18i)37-s + 3.93i·41-s + (−0.659 − 0.659i)43-s + (−9.43 − 9.43i)47-s + ⋯
L(s)  = 1  + (0.801 + 0.598i)5-s + (−0.784 + 0.784i)7-s + 0.972i·11-s + (1.36 + 1.36i)13-s + (0.834 + 0.834i)17-s − 0.250i·19-s + (−0.147 + 0.147i)23-s + (0.283 + 0.958i)25-s + 1.19·29-s − 0.912·31-s + (−1.09 + 0.159i)35-s + (0.523 − 0.523i)37-s + 0.613i·41-s + (−0.100 − 0.100i)43-s + (−1.37 − 1.37i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.518 - 0.854i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.518 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.090973549\)
\(L(\frac12)\) \(\approx\) \(2.090973549\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 - 1.33i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (2.07 - 2.07i)T - 7iT^{2} \)
11 \( 1 - 3.22iT - 11T^{2} \)
13 \( 1 + (-4.92 - 4.92i)T + 13iT^{2} \)
17 \( 1 + (-3.44 - 3.44i)T + 17iT^{2} \)
19 \( 1 + 1.09iT - 19T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + 5.08T + 31T^{2} \)
37 \( 1 + (-3.18 + 3.18i)T - 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (0.659 + 0.659i)T + 43iT^{2} \)
47 \( 1 + (9.43 + 9.43i)T + 47iT^{2} \)
53 \( 1 + (-5.54 + 5.54i)T - 53iT^{2} \)
59 \( 1 + 2.11T + 59T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + (-7.76 + 7.76i)T - 67iT^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 + (7.21 + 7.21i)T + 73iT^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + (0.143 - 0.143i)T - 83iT^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 + (-5.05 + 5.05i)T - 97iT^{2} \)
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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

−8.855681768756716922222835294126, −7.977332859488722305734619613963, −6.89954698376350079433642314530, −6.47058672054737085716220368000, −5.91164301266350204632002241975, −5.06396708094290984169965056772, −3.95631884379632085998180055852, −3.24073867020359589442729304084, −2.20190840998276151025068959200, −1.52611734482387296242886685433, 0.64182761555282734604209381723, 1.26158194782960400512304199447, 2.88645167476048790487367879100, 3.36517427103453563132021043314, 4.35654970867065023924053399006, 5.44110531001176620857161705941, 5.88587700472288831328941136216, 6.53146099895526807768624468144, 7.49878927994491004822643881324, 8.328524828701805592315668645583

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