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Properties

Label 2-4140-5.4-c1-0-42
Degree $2$
Conductor $4140$
Sign $-0.276 + 0.960i$
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  + (−2.14 − 0.619i)5-s + 1.66i·7-s + 5.96·11-s − 3.02i·13-s − 6.90i·17-s − 6.93·19-s + i·23-s + (4.23 + 2.66i)25-s + 7.66·29-s − 5.56·31-s + (1.02 − 3.56i)35-s + 6.17i·37-s − 6.47·41-s + 3.84i·43-s + 7.29i·47-s + ⋯
L(s)  = 1  + (−0.960 − 0.276i)5-s + 0.627i·7-s + 1.79·11-s − 0.840i·13-s − 1.67i·17-s − 1.58·19-s + 0.208i·23-s + (0.846 + 0.532i)25-s + 1.42·29-s − 0.999·31-s + (0.173 − 0.603i)35-s + 1.01i·37-s − 1.01·41-s + 0.586i·43-s + 1.06i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.276 + 0.960i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.276 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082233768\)
\(L(\frac12)\) \(\approx\) \(1.082233768\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 + 0.619i)T \)
23 \( 1 - iT \)
good7 \( 1 - 1.66iT - 7T^{2} \)
11 \( 1 - 5.96T + 11T^{2} \)
13 \( 1 + 3.02iT - 13T^{2} \)
17 \( 1 + 6.90iT - 17T^{2} \)
19 \( 1 + 6.93T + 19T^{2} \)
29 \( 1 - 7.66T + 29T^{2} \)
31 \( 1 + 5.56T + 31T^{2} \)
37 \( 1 - 6.17iT - 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 - 3.84iT - 43T^{2} \)
47 \( 1 - 7.29iT - 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 + 5.53T + 59T^{2} \)
61 \( 1 - 1.81T + 61T^{2} \)
67 \( 1 + 9.32iT - 67T^{2} \)
71 \( 1 + 9.14T + 71T^{2} \)
73 \( 1 + 8.35iT - 73T^{2} \)
79 \( 1 + 3.89T + 79T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 - 8.30T + 89T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−8.336892352318521922475812899299, −7.49087490666965711688174289925, −6.71081935794760318810672381622, −6.15562059544646955353182651236, −4.99857400454008153729299888984, −4.50356716018809004289089989816, −3.55009025815331843178683518053, −2.83548067293038111201692645821, −1.52748216378153422987075035023, −0.34646706041899106381138944222, 1.15574012433728430673676742723, 2.14971359345608055860161435977, 3.60237719912454587009708137227, 4.06753511549251297960007827234, 4.43584893450331595421900326867, 5.90907459837211411958630518940, 6.70847697779939747447521386207, 6.91472384252217933659816430286, 7.941255277907946361649885709564, 8.809350181260015414942462604656

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