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Properties

Label 2-1512-24.11-c1-0-27
Degree $2$
Conductor $1512$
Sign $0.643 - 0.765i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−0.971 + 1.02i)2-s + (−0.114 − 1.99i)4-s − 2.21·5-s − i·7-s + (2.16 + 1.82i)8-s + (2.14 − 2.27i)10-s + 2.42i·11-s − 1.11i·13-s + (1.02 + 0.971i)14-s + (−3.97 + 0.456i)16-s + 0.701i·17-s + 0.938·19-s + (0.252 + 4.41i)20-s + (−2.49 − 2.35i)22-s − 4.30·23-s + ⋯
L(s)  = 1  + (−0.686 + 0.727i)2-s + (−0.0571 − 0.998i)4-s − 0.989·5-s − 0.377i·7-s + (0.765 + 0.643i)8-s + (0.679 − 0.719i)10-s + 0.730i·11-s − 0.309i·13-s + (0.274 + 0.259i)14-s + (−0.993 + 0.114i)16-s + 0.170i·17-s + 0.215·19-s + (0.0565 + 0.988i)20-s + (−0.531 − 0.501i)22-s − 0.897·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8119716526\)
\(L(\frac12)\) \(\approx\) \(0.8119716526\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.971 - 1.02i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 2.21T + 5T^{2} \)
11 \( 1 - 2.42iT - 11T^{2} \)
13 \( 1 + 1.11iT - 13T^{2} \)
17 \( 1 - 0.701iT - 17T^{2} \)
19 \( 1 - 0.938T + 19T^{2} \)
23 \( 1 + 4.30T + 23T^{2} \)
29 \( 1 - 4.26T + 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 - 3.12iT - 37T^{2} \)
41 \( 1 - 0.157iT - 41T^{2} \)
43 \( 1 - 7.08T + 43T^{2} \)
47 \( 1 - 0.867T + 47T^{2} \)
53 \( 1 - 3.91T + 53T^{2} \)
59 \( 1 - 7.28iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 - 7.02T + 73T^{2} \)
79 \( 1 - 6.65iT - 79T^{2} \)
83 \( 1 - 8.41iT - 83T^{2} \)
89 \( 1 + 7.34iT - 89T^{2} \)
97 \( 1 - 7.99T + 97T^{2} \)
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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

−9.611944411120985830581170918376, −8.555227636774707730964474001103, −7.87932446660324461274645607680, −7.40630356332985214336040749393, −6.54688381090021529556178915250, −5.62787784847353470341560312633, −4.56540765994767510785415905193, −3.86161238350578775935174600881, −2.27371922909222167280245941057, −0.76176761961149148208862314111, 0.64593927671029797945305773287, 2.13777229278982915278875351092, 3.27159657453752124644493340279, 3.95017016179582966874980482740, 4.99104605571893110149512328398, 6.27139042866479773134685501887, 7.23247833709022779046239066646, 8.022221328778638476402182381402, 8.556330818461148425634770942339, 9.319450660707963561977915760357

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