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Properties

Label 2-1512-24.11-c1-0-52
Degree $2$
Conductor $1512$
Sign $0.975 - 0.218i$
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  + (−1.16 + 0.795i)2-s + (0.735 − 1.85i)4-s + 2.09·5-s + i·7-s + (0.618 + 2.75i)8-s + (−2.44 + 1.66i)10-s − 0.961i·11-s − 4.60i·13-s + (−0.795 − 1.16i)14-s + (−2.91 − 2.73i)16-s + 3.28i·17-s + 3.66·19-s + (1.53 − 3.89i)20-s + (0.764 + 1.12i)22-s + 2.40·23-s + ⋯
L(s)  = 1  + (−0.826 + 0.562i)2-s + (0.367 − 0.929i)4-s + 0.936·5-s + 0.377i·7-s + (0.218 + 0.975i)8-s + (−0.774 + 0.526i)10-s − 0.289i·11-s − 1.27i·13-s + (−0.212 − 0.312i)14-s + (−0.729 − 0.684i)16-s + 0.795i·17-s + 0.841·19-s + (0.344 − 0.870i)20-s + (0.162 + 0.239i)22-s + 0.501·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.975 - 0.218i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.393181941\)
\(L(\frac12)\) \(\approx\) \(1.393181941\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.795i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 2.09T + 5T^{2} \)
11 \( 1 + 0.961iT - 11T^{2} \)
13 \( 1 + 4.60iT - 13T^{2} \)
17 \( 1 - 3.28iT - 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 - 9.31T + 29T^{2} \)
31 \( 1 + 1.84iT - 31T^{2} \)
37 \( 1 + 2.29iT - 37T^{2} \)
41 \( 1 + 0.314iT - 41T^{2} \)
43 \( 1 + 8.64T + 43T^{2} \)
47 \( 1 - 2.34T + 47T^{2} \)
53 \( 1 - 7.73T + 53T^{2} \)
59 \( 1 - 7.99iT - 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 - 4.12T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 1.26iT - 79T^{2} \)
83 \( 1 + 2.99iT - 83T^{2} \)
89 \( 1 + 12.8iT - 89T^{2} \)
97 \( 1 - 4.98T + 97T^{2} \)
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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

−9.518011784924686673235859960368, −8.604075194332748428164309726646, −8.097554956863645484083329154194, −7.12138866218462831703878313294, −6.17156681132705540196142937231, −5.66629085075646329791179150253, −4.90186852402052414070258362300, −3.19340475494134598754696195773, −2.13787721705668225779469746904, −0.908038369051725368148527626751, 1.08693534467430454792327409676, 2.11599759047505861848921402951, 3.08368478077031880117889028594, 4.27381085817820439392030571403, 5.23060162042837062664024203210, 6.61725986990426560424219450969, 6.94111917639305855439055483618, 8.002801695830369208275060715922, 8.897622804732399878967724407149, 9.572874302126485828422919089293

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