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Properties

Label 2-4140-5.4-c1-0-35
Degree $2$
Conductor $4140$
Sign $0.583 - 0.812i$
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  + (1.81 + 1.30i)5-s + 3.73i·7-s + 5.75·11-s − 4.85i·13-s + 5.12i·17-s + 8.11·19-s + i·23-s + (1.59 + 4.73i)25-s − 0.516·29-s + 6.47·31-s + (−4.87 + 6.78i)35-s − 7.07i·37-s − 4.42·41-s + 1.54i·43-s − 10.0i·47-s + ⋯
L(s)  = 1  + (0.812 + 0.583i)5-s + 1.41i·7-s + 1.73·11-s − 1.34i·13-s + 1.24i·17-s + 1.86·19-s + 0.208i·23-s + (0.319 + 0.947i)25-s − 0.0958·29-s + 1.16·31-s + (−0.824 + 1.14i)35-s − 1.16i·37-s − 0.691·41-s + 0.235i·43-s − 1.47i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.583 - 0.812i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.583 - 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.829874610\)
\(L(\frac12)\) \(\approx\) \(2.829874610\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 - 1.30i)T \)
23 \( 1 - iT \)
good7 \( 1 - 3.73iT - 7T^{2} \)
11 \( 1 - 5.75T + 11T^{2} \)
13 \( 1 + 4.85iT - 13T^{2} \)
17 \( 1 - 5.12iT - 17T^{2} \)
19 \( 1 - 8.11T + 19T^{2} \)
29 \( 1 + 0.516T + 29T^{2} \)
31 \( 1 - 6.47T + 31T^{2} \)
37 \( 1 + 7.07iT - 37T^{2} \)
41 \( 1 + 4.42T + 41T^{2} \)
43 \( 1 - 1.54iT - 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 - 2.22iT - 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 8.86iT - 67T^{2} \)
71 \( 1 - 5.93T + 71T^{2} \)
73 \( 1 + 14.3iT - 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 - 3.73iT - 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 0.693iT - 97T^{2} \)
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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

−8.651605407032930001060595518359, −7.85793514243845412804025589381, −6.92261209842384807314947669692, −6.15565935800356324582518334039, −5.72142870940399268305555258968, −5.06901253734455282976335003047, −3.67443052102079640481278980406, −3.10270662892262176509860592760, −2.11467766917683280542617953416, −1.21087682997162227375812450039, 1.03364269239101895388846185889, 1.38364896151068049490584350300, 2.81960358924368526212472963063, 3.89484479544844335311368806157, 4.49305244507578896682503300106, 5.18724390196701290870253623017, 6.32090834656937523344578821418, 6.83432721929098802148840705096, 7.35397221896275946366188479982, 8.429969219828102781219955092933

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