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Properties

Label 2-1512-24.11-c1-0-64
Degree $2$
Conductor $1512$
Sign $0.753 + 0.657i$
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  + (1.37 + 0.334i)2-s + (1.77 + 0.920i)4-s − 4.18·5-s − i·7-s + (2.13 + 1.85i)8-s + (−5.74 − 1.40i)10-s − 1.86i·11-s − 3.07i·13-s + (0.334 − 1.37i)14-s + (2.30 + 3.26i)16-s + 0.504i·17-s + 3.05·19-s + (−7.42 − 3.84i)20-s + (0.625 − 2.56i)22-s + 5.97·23-s + ⋯
L(s)  = 1  + (0.971 + 0.236i)2-s + (0.887 + 0.460i)4-s − 1.87·5-s − 0.377i·7-s + (0.753 + 0.657i)8-s + (−1.81 − 0.443i)10-s − 0.563i·11-s − 0.853i·13-s + (0.0895 − 0.367i)14-s + (0.576 + 0.817i)16-s + 0.122i·17-s + 0.701·19-s + (−1.66 − 0.860i)20-s + (0.133 − 0.546i)22-s + 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.753 + 0.657i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.111371686\)
\(L(\frac12)\) \(\approx\) \(2.111371686\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.334i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 4.18T + 5T^{2} \)
11 \( 1 + 1.86iT - 11T^{2} \)
13 \( 1 + 3.07iT - 13T^{2} \)
17 \( 1 - 0.504iT - 17T^{2} \)
19 \( 1 - 3.05T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 + 3.52T + 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + 7.26iT - 41T^{2} \)
43 \( 1 - 9.02T + 43T^{2} \)
47 \( 1 + 0.327T + 47T^{2} \)
53 \( 1 + 8.32T + 53T^{2} \)
59 \( 1 + 5.98iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 + 9.21T + 67T^{2} \)
71 \( 1 - 9.02T + 71T^{2} \)
73 \( 1 - 0.416T + 73T^{2} \)
79 \( 1 - 6.00iT - 79T^{2} \)
83 \( 1 - 2.62iT - 83T^{2} \)
89 \( 1 - 5.69iT - 89T^{2} \)
97 \( 1 - 6.21T + 97T^{2} \)
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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

−9.185179536970669272258058574808, −8.217156052535388195301701543121, −7.48601097587276935711618273418, −7.27998071150710335309234331227, −5.97294392940608646547853246033, −5.14240055487850619481499781690, −4.09605133633464068606524517339, −3.64404245955252785863856810020, −2.72746860448881720991250744019, −0.67072368945154268041714121299, 1.32614156356895743505216479293, 2.93881104343517683892880446127, 3.53220988308914161941966113016, 4.68154712711097911171275928604, 4.89862044517774571926091888954, 6.40849433509978441918116116500, 7.12075183769499899828998233530, 7.69642220934007921845383992867, 8.689943723625467169682724326418, 9.596559601875504845858533649322

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