Properties

Label 2-4140-5.4-c1-0-26
Degree $2$
Conductor $4140$
Sign $0.593 - 0.804i$
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 + 1.32i)5-s + 0.476i·7-s + 3.39·11-s − 2.28i·13-s + 2.61i·17-s − 5.08·19-s i·23-s + (1.47 + 4.77i)25-s + 3.19·29-s + 3.65·31-s + (−0.632 + 0.857i)35-s + 8.44i·37-s + 5.25·41-s + 0.269i·43-s + 8.26i·47-s + ⋯
L(s)  = 1  + (0.804 + 0.593i)5-s + 0.180i·7-s + 1.02·11-s − 0.634i·13-s + 0.634i·17-s − 1.16·19-s − 0.208i·23-s + (0.295 + 0.955i)25-s + 0.592·29-s + 0.655·31-s + (−0.106 + 0.145i)35-s + 1.38i·37-s + 0.820·41-s + 0.0410i·43-s + 1.20i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.593 - 0.804i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.593 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.342134458\)
\(L(\frac12)\) \(\approx\) \(2.342134458\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 - 1.32i)T \)
23 \( 1 + iT \)
good7 \( 1 - 0.476iT - 7T^{2} \)
11 \( 1 - 3.39T + 11T^{2} \)
13 \( 1 + 2.28iT - 13T^{2} \)
17 \( 1 - 2.61iT - 17T^{2} \)
19 \( 1 + 5.08T + 19T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 - 8.44iT - 37T^{2} \)
41 \( 1 - 5.25T + 41T^{2} \)
43 \( 1 - 0.269iT - 43T^{2} \)
47 \( 1 - 8.26iT - 47T^{2} \)
53 \( 1 + 7.77iT - 53T^{2} \)
59 \( 1 + 8.35T + 59T^{2} \)
61 \( 1 - 5.45T + 61T^{2} \)
67 \( 1 + 6.83iT - 67T^{2} \)
71 \( 1 - 6.28T + 71T^{2} \)
73 \( 1 - 4.90iT - 73T^{2} \)
79 \( 1 - 9.29T + 79T^{2} \)
83 \( 1 - 6.19iT - 83T^{2} \)
89 \( 1 - 0.423T + 89T^{2} \)
97 \( 1 - 8.67iT - 97T^{2} \)
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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

−8.498223131228870340926444103979, −7.916921067845246343811420142164, −6.74469798420640286047047674773, −6.42677940332564067759819492593, −5.75930846047598638346983342320, −4.78336377835813125113890127717, −3.92239414939233834870044169541, −2.97873843206486480271709557734, −2.16310327698913289996960232989, −1.12922781331145911543954478605, 0.74394207488774772272624346599, 1.79208822442681414770346828480, 2.60572780682595330732985825177, 3.94354388534606140166336896779, 4.45325967082583426572503742167, 5.35876850039820174270823146669, 6.18768553666958412414303165675, 6.69982796830683601701171707300, 7.52716864531599174067812951120, 8.555684850474281133186363566149

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