Properties

Degree $2$
Conductor $1512$
Sign $-0.652 - 0.757i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 3.45i)5-s + (−0.657 − 2.56i)7-s + (0.444 − 0.769i)11-s − 6.98·13-s + (0.342 − 0.592i)17-s + (1.73 + 3.00i)19-s + (−2.94 − 5.10i)23-s + (−5.43 + 9.42i)25-s + 5.87·29-s + (−0.604 + 1.04i)31-s + (−7.53 + 7.37i)35-s + (4.32 + 7.49i)37-s + 6.30·41-s + 0.888·43-s + (−1.59 − 2.76i)47-s + ⋯
L(s)  = 1  + (−0.890 − 1.54i)5-s + (−0.248 − 0.968i)7-s + (0.133 − 0.231i)11-s − 1.93·13-s + (0.0830 − 0.143i)17-s + (0.397 + 0.688i)19-s + (−0.614 − 1.06i)23-s + (−1.08 + 1.88i)25-s + 1.09·29-s + (−0.108 + 0.187i)31-s + (−1.27 + 1.24i)35-s + (0.711 + 1.23i)37-s + 0.983·41-s + 0.135·43-s + (−0.232 − 0.402i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.652 - 0.757i$
Motivic weight: \(1\)
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.652 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2949557460\)
\(L(\frac12)\) \(\approx\) \(0.2949557460\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.657 + 2.56i)T \)
good5 \( 1 + (1.99 + 3.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.444 + 0.769i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.98T + 13T^{2} \)
17 \( 1 + (-0.342 + 0.592i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 - 3.00i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.94 + 5.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.87T + 29T^{2} \)
31 \( 1 + (0.604 - 1.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.32 - 7.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.30T + 41T^{2} \)
43 \( 1 - 0.888T + 43T^{2} \)
47 \( 1 + (1.59 + 2.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.890 + 1.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.11 - 8.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.882 - 1.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.83 - 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.50T + 71T^{2} \)
73 \( 1 + (-5.14 + 8.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.64 + 4.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (5.77 + 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−8.916649167100178646209048448471, −8.032049044437303365067605475107, −7.60824427465047620240998401347, −6.66817208025957658794382089853, −5.42115225616778848165726466019, −4.54383616832750668191620952871, −4.17230574901160942437855207238, −2.84764526198768120201437054545, −1.19107242590372255888706191673, −0.12559548770747369883499609350, 2.37232114338093050156854234112, 2.85873634737547917703270796811, 3.94264273614852086687627037531, 5.00539347100485724628028820008, 6.03138593284635272626314916372, 6.91689676884829150529785914227, 7.48547988087112296745921811033, 8.145717945452111449259642400416, 9.530124245190455547350373801986, 9.742401524989987466941491239639

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