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Properties

Label 2-1512-24.11-c1-0-9
Degree $2$
Conductor $1512$
Sign $0.171 - 0.985i$
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  + (0.0811 − 1.41i)2-s + (−1.98 − 0.229i)4-s + 1.27·5-s + i·7-s + (−0.485 + 2.78i)8-s + (0.103 − 1.80i)10-s + 2.84i·11-s + 0.223i·13-s + (1.41 + 0.0811i)14-s + (3.89 + 0.911i)16-s + 0.397i·17-s − 4.64·19-s + (−2.53 − 0.293i)20-s + (4.02 + 0.231i)22-s − 7.36·23-s + ⋯
L(s)  = 1  + (0.0574 − 0.998i)2-s + (−0.993 − 0.114i)4-s + 0.571·5-s + 0.377i·7-s + (−0.171 + 0.985i)8-s + (0.0328 − 0.570i)10-s + 0.858i·11-s + 0.0620i·13-s + (0.377 + 0.0217i)14-s + (0.973 + 0.227i)16-s + 0.0964i·17-s − 1.06·19-s + (−0.567 − 0.0655i)20-s + (0.857 + 0.0493i)22-s − 1.53·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.171 - 0.985i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.171 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6155011135\)
\(L(\frac12)\) \(\approx\) \(0.6155011135\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0811 + 1.41i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 1.27T + 5T^{2} \)
11 \( 1 - 2.84iT - 11T^{2} \)
13 \( 1 - 0.223iT - 13T^{2} \)
17 \( 1 - 0.397iT - 17T^{2} \)
19 \( 1 + 4.64T + 19T^{2} \)
23 \( 1 + 7.36T + 23T^{2} \)
29 \( 1 + 10.6T + 29T^{2} \)
31 \( 1 - 7.67iT - 31T^{2} \)
37 \( 1 + 4.74iT - 37T^{2} \)
41 \( 1 + 1.97iT - 41T^{2} \)
43 \( 1 - 7.62T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 0.295T + 53T^{2} \)
59 \( 1 - 7.25iT - 59T^{2} \)
61 \( 1 - 9.45iT - 61T^{2} \)
67 \( 1 + 3.52T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 5.22iT - 79T^{2} \)
83 \( 1 + 9.11iT - 83T^{2} \)
89 \( 1 + 8.94iT - 89T^{2} \)
97 \( 1 - 0.228T + 97T^{2} \)
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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

−9.723963839799424093781921154359, −9.125524233749553899989417407887, −8.297475851385047606122997673737, −7.36340909920636786252890370232, −6.11906044503031860568581159889, −5.46990169149928980380595359859, −4.42746821912890995180406920492, −3.65209627138333665312472967120, −2.27248593320878622506327157154, −1.77837857808980231543588405928, 0.22297095184429159954417645251, 1.98049577111915551635479668596, 3.56254893735066714957393818999, 4.27505104454386676700072856803, 5.44655022125560804105618164389, 6.06297497095090913156229982991, 6.68140744804292678412932980389, 7.938038818987167850813355718592, 8.110423322658556646809836961818, 9.400000095362510430451310868677

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