Properties

Label 2-4140-345.344-c1-0-27
Degree $2$
Conductor $4140$
Sign $0.924 - 0.380i$
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  + (2.23 + 0.114i)5-s + 3.07·7-s + 1.12·11-s + 3.92i·13-s + 0.286i·17-s − 5.12i·19-s + (4.17 + 2.35i)23-s + (4.97 + 0.510i)25-s + 3.70i·29-s − 5.14·31-s + (6.86 + 0.351i)35-s + 1.46·37-s + 4.70i·41-s + 4.70·43-s + 3.27·47-s + ⋯
L(s)  = 1  + (0.998 + 0.0511i)5-s + 1.16·7-s + 0.338·11-s + 1.08i·13-s + 0.0693i·17-s − 1.17i·19-s + (0.870 + 0.491i)23-s + (0.994 + 0.102i)25-s + 0.688i·29-s − 0.923·31-s + (1.16 + 0.0593i)35-s + 0.240·37-s + 0.734i·41-s + 0.716·43-s + 0.477·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.924 - 0.380i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.924 - 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.929392897\)
\(L(\frac12)\) \(\approx\) \(2.929392897\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.114i)T \)
23 \( 1 + (-4.17 - 2.35i)T \)
good7 \( 1 - 3.07T + 7T^{2} \)
11 \( 1 - 1.12T + 11T^{2} \)
13 \( 1 - 3.92iT - 13T^{2} \)
17 \( 1 - 0.286iT - 17T^{2} \)
19 \( 1 + 5.12iT - 19T^{2} \)
29 \( 1 - 3.70iT - 29T^{2} \)
31 \( 1 + 5.14T + 31T^{2} \)
37 \( 1 - 1.46T + 37T^{2} \)
41 \( 1 - 4.70iT - 41T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 - 3.27T + 47T^{2} \)
53 \( 1 + 3.68iT - 53T^{2} \)
59 \( 1 + 1.30iT - 59T^{2} \)
61 \( 1 - 5.75iT - 61T^{2} \)
67 \( 1 - 7.13T + 67T^{2} \)
71 \( 1 + 1.29iT - 71T^{2} \)
73 \( 1 + 9.83iT - 73T^{2} \)
79 \( 1 + 0.954iT - 79T^{2} \)
83 \( 1 - 7.87iT - 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.711324308880131304589638810752, −7.64178877720462730824021027911, −6.95818607777665248911399312484, −6.34223690610684184026103422274, −5.32039638393916942344495807605, −4.89152221807416954595018129778, −4.01179860345848288284191339735, −2.80953698606512765776110299937, −1.91486913072087990355959136857, −1.18954836409057887221366970362, 0.957638881338281579659888745044, 1.85207298797753558111916247446, 2.71809238874586231685400854591, 3.81911028636814250660275654361, 4.75561593304619449728973723644, 5.50458911734665906221536539981, 5.93444737146695211884905355119, 6.94351286759393802568309254445, 7.73239982291044262584384637411, 8.364448995661240692743240418848

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