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Properties

Label 2-4140-5.4-c1-0-31
Degree $2$
Conductor $4140$
Sign $0.935 + 0.354i$
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  + (−0.792 + 2.09i)5-s − 4.31i·7-s + 2.26·11-s + 2.25i·13-s + 0.535i·17-s + 6.91·19-s + i·23-s + (−3.74 − 3.31i)25-s + 5.64·29-s − 10.4·31-s + (9.02 + 3.42i)35-s − 8.10i·37-s − 0.633·41-s − 1.30i·43-s + 12.4i·47-s + ⋯
L(s)  = 1  + (−0.354 + 0.935i)5-s − 1.63i·7-s + 0.683·11-s + 0.625i·13-s + 0.129i·17-s + 1.58·19-s + 0.208i·23-s + (−0.748 − 0.663i)25-s + 1.04·29-s − 1.87·31-s + (1.52 + 0.578i)35-s − 1.33i·37-s − 0.0988·41-s − 0.199i·43-s + 1.81i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.935 + 0.354i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.935 + 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.823900106\)
\(L(\frac12)\) \(\approx\) \(1.823900106\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.792 - 2.09i)T \)
23 \( 1 - iT \)
good7 \( 1 + 4.31iT - 7T^{2} \)
11 \( 1 - 2.26T + 11T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 - 0.535iT - 17T^{2} \)
19 \( 1 - 6.91T + 19T^{2} \)
29 \( 1 - 5.64T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 8.10iT - 37T^{2} \)
41 \( 1 + 0.633T + 41T^{2} \)
43 \( 1 + 1.30iT - 43T^{2} \)
47 \( 1 - 12.4iT - 47T^{2} \)
53 \( 1 + 5.98iT - 53T^{2} \)
59 \( 1 - 7.01T + 59T^{2} \)
61 \( 1 - 5.98T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 + 0.151T + 71T^{2} \)
73 \( 1 - 8.88iT - 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 0.716iT - 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 + 16.2iT - 97T^{2} \)
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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

−8.160182540881295672634209860624, −7.36977800523174108360424530970, −7.11496788693089480544833993465, −6.42312129689421344585289805692, −5.42683372731472501937950999602, −4.32842402593860610805155560805, −3.77640122635294487546811632253, −3.15326701179532727706780230683, −1.81260829243582352283617320952, −0.69905580451837754350248584009, 0.893331236530548225091146721308, 1.97306693377183277912503359569, 3.03558385952370617201914639660, 3.80355340568078874853206764187, 5.07028474368975610297637412717, 5.27088591278929867281710627371, 6.09685134897714410719854798316, 7.03251421533338570906166602709, 7.945006235102419685229293526181, 8.504451425015791688563547370154

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