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Properties

Label 2-4140-5.4-c1-0-41
Degree $2$
Conductor $4140$
Sign $0.344 + 0.938i$
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  + (2.09 − 0.770i)5-s + 2.34i·7-s − 4.15·11-s − 5.10i·13-s − 0.0253i·17-s + 0.536·19-s + i·23-s + (3.81 − 3.23i)25-s + 9.72·29-s − 8.15·31-s + (1.80 + 4.91i)35-s + 2.50i·37-s − 4.20·41-s − 11.5i·43-s + 1.22i·47-s + ⋯
L(s)  = 1  + (0.938 − 0.344i)5-s + 0.884i·7-s − 1.25·11-s − 1.41i·13-s − 0.00615i·17-s + 0.123·19-s + 0.208i·23-s + (0.762 − 0.647i)25-s + 1.80·29-s − 1.46·31-s + (0.304 + 0.830i)35-s + 0.411i·37-s − 0.656·41-s − 1.75i·43-s + 0.178i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.344 + 0.938i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.786880324\)
\(L(\frac12)\) \(\approx\) \(1.786880324\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 0.770i)T \)
23 \( 1 - iT \)
good7 \( 1 - 2.34iT - 7T^{2} \)
11 \( 1 + 4.15T + 11T^{2} \)
13 \( 1 + 5.10iT - 13T^{2} \)
17 \( 1 + 0.0253iT - 17T^{2} \)
19 \( 1 - 0.536T + 19T^{2} \)
29 \( 1 - 9.72T + 29T^{2} \)
31 \( 1 + 8.15T + 31T^{2} \)
37 \( 1 - 2.50iT - 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 - 1.22iT - 47T^{2} \)
53 \( 1 + 14.3iT - 53T^{2} \)
59 \( 1 - 4.95T + 59T^{2} \)
61 \( 1 - 8.57T + 61T^{2} \)
67 \( 1 + 8.41iT - 67T^{2} \)
71 \( 1 + 1.22T + 71T^{2} \)
73 \( 1 + 3.30iT - 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 - 3.55iT - 83T^{2} \)
89 \( 1 - 6.97T + 89T^{2} \)
97 \( 1 + 9.13iT - 97T^{2} \)
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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

−8.505244646099823393805058318885, −7.64835274218766076696801133944, −6.74496576245405499218160859782, −5.80434371661806423370561572866, −5.34842530718887559660767223920, −4.90949742557379616078005851009, −3.42972162915344781738424637038, −2.65239189353104903058558931401, −1.93599453359651084686490450204, −0.52103348038823352562636695557, 1.15110635300132788746166601585, 2.20612599481254440151526889060, 2.96354322621623960715532643966, 4.08195009757407812012039051657, 4.81941900161618297364863872226, 5.61705722180485515067192201093, 6.46480284136905646715296981334, 7.04466978736452982225668925846, 7.68902627822491541557223997255, 8.637438589007094958902439153470

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