Properties

Label 2-4140-5.4-c1-0-18
Degree $2$
Conductor $4140$
Sign $0.992 - 0.122i$
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  + (−0.274 − 2.21i)5-s − 0.615i·7-s − 5.98·11-s + 1.31i·13-s + 5.22i·17-s + 6.47·19-s + i·23-s + (−4.84 + 1.21i)25-s − 3.81·29-s + 2.89·31-s + (−1.36 + 0.168i)35-s − 3.05i·37-s + 8.67·41-s − 2.78i·43-s + 6.52i·47-s + ⋯
L(s)  = 1  + (−0.122 − 0.992i)5-s − 0.232i·7-s − 1.80·11-s + 0.363i·13-s + 1.26i·17-s + 1.48·19-s + 0.208i·23-s + (−0.969 + 0.243i)25-s − 0.707·29-s + 0.520·31-s + (−0.230 + 0.0285i)35-s − 0.502i·37-s + 1.35·41-s − 0.425i·43-s + 0.952i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.992 - 0.122i$
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: \((2,\ 4140,\ (\ :1/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418929458\)
\(L(\frac12)\) \(\approx\) \(1.418929458\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.274 + 2.21i)T \)
23 \( 1 - iT \)
good7 \( 1 + 0.615iT - 7T^{2} \)
11 \( 1 + 5.98T + 11T^{2} \)
13 \( 1 - 1.31iT - 13T^{2} \)
17 \( 1 - 5.22iT - 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
29 \( 1 + 3.81T + 29T^{2} \)
31 \( 1 - 2.89T + 31T^{2} \)
37 \( 1 + 3.05iT - 37T^{2} \)
41 \( 1 - 8.67T + 41T^{2} \)
43 \( 1 + 2.78iT - 43T^{2} \)
47 \( 1 - 6.52iT - 47T^{2} \)
53 \( 1 - 8.11iT - 53T^{2} \)
59 \( 1 + 8.93T + 59T^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 - 15.0T + 71T^{2} \)
73 \( 1 + 16.1iT - 73T^{2} \)
79 \( 1 - 1.59T + 79T^{2} \)
83 \( 1 - 7.29iT - 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−8.249460053709770079857057372493, −7.80301544220234974894732779958, −7.24538424226029362352674979508, −5.91834655820969250069431479765, −5.52512599952681445070531442827, −4.69948854567293010124658288771, −3.96641293856217719734330967762, −2.95783948547140838656824338252, −1.90605128093553497208077867162, −0.794061410162024236443317665420, 0.55451446842482385777166395572, 2.29448230698426217065262368126, 2.85376488432096204190763422579, 3.54706192192137768188474328482, 4.87820450380326963549164242838, 5.36720005857600680646689997117, 6.16867288210316269578841802067, 7.16177789464775555119872863540, 7.61509155753236741548417090919, 8.136929958004775194406858328404

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