Properties

Degree $2$
Conductor $1512$
Sign $-0.997 + 0.0754i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 − 2.95i)5-s + (−2.27 + 1.35i)7-s + (−3.01 − 5.22i)11-s + 0.417·13-s + (−1.27 − 2.20i)17-s + (−1.74 + 3.01i)19-s + (−4.54 + 7.87i)23-s + (−3.33 − 5.78i)25-s + 1.67·29-s + (−3.82 − 6.62i)31-s + (0.110 + 9.04i)35-s + (−4.69 + 8.12i)37-s + 2.13·41-s − 6.03·43-s + (3.94 − 6.84i)47-s + ⋯
L(s)  = 1  + (0.764 − 1.32i)5-s + (−0.859 + 0.510i)7-s + (−0.909 − 1.57i)11-s + 0.115·13-s + (−0.309 − 0.535i)17-s + (−0.399 + 0.692i)19-s + (−0.948 + 1.64i)23-s + (−0.667 − 1.15i)25-s + 0.311·29-s + (−0.686 − 1.18i)31-s + (0.0186 + 1.52i)35-s + (−0.771 + 1.33i)37-s + 0.333·41-s − 0.919·43-s + (0.576 − 0.997i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.997 + 0.0754i$
Motivic weight: \(1\)
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.997 + 0.0754i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6203199929\)
\(L(\frac12)\) \(\approx\) \(0.6203199929\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.27 - 1.35i)T \)
good5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.01 + 5.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.417T + 13T^{2} \)
17 \( 1 + (1.27 + 2.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.74 - 3.01i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.54 - 7.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.67T + 29T^{2} \)
31 \( 1 + (3.82 + 6.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.69 - 8.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.13T + 41T^{2} \)
43 \( 1 + 6.03T + 43T^{2} \)
47 \( 1 + (-3.94 + 6.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.967 + 1.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.91 - 6.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.67 - 8.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.516 + 0.894i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.650T + 71T^{2} \)
73 \( 1 + (0.642 + 1.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.20 + 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + (-6.70 + 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.26T + 97T^{2} \)
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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

−8.995359368152889110553227346544, −8.516453498023489965822773915225, −7.68033497126042613916671597530, −6.31660562347596838885130677268, −5.66677134789823053962403345543, −5.25905264037173352245813974694, −3.90696200893927589429050960213, −2.89910646818735846884281196447, −1.68315398748542842305828104739, −0.22216158605547059221214188696, 2.07561076705768704329819370517, 2.69524914439081703605454820465, 3.84660062345302518684917080427, 4.85513545858554057275998798979, 6.01558906474582250496195956099, 6.80074078960148637899870321048, 7.08522704598783947286293927823, 8.182577693619517208232624030413, 9.338679233036082193813908806591, 9.991039276481970169580685571744

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