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Properties

Label 2-1512-21.5-c1-0-18
Degree $2$
Conductor $1512$
Sign $0.991 + 0.131i$
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  + (0.779 − 1.35i)5-s + (2.28 + 1.33i)7-s + (−4.09 + 2.36i)11-s − 5.16i·13-s + (1.33 + 2.31i)17-s + (3.08 + 1.77i)19-s + (7.21 + 4.16i)23-s + (1.28 + 2.22i)25-s − 3.56i·29-s + (7.47 − 4.31i)31-s + (3.58 − 2.04i)35-s + (3.31 − 5.73i)37-s − 2.84·41-s − 6.78·43-s + (2.84 − 4.93i)47-s + ⋯
L(s)  = 1  + (0.348 − 0.603i)5-s + (0.863 + 0.504i)7-s + (−1.23 + 0.713i)11-s − 1.43i·13-s + (0.323 + 0.560i)17-s + (0.707 + 0.408i)19-s + (1.50 + 0.868i)23-s + (0.256 + 0.444i)25-s − 0.662i·29-s + (1.34 − 0.775i)31-s + (0.605 − 0.345i)35-s + (0.544 − 0.943i)37-s − 0.444·41-s − 1.03·43-s + (0.415 − 0.719i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.991 + 0.131i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.991 + 0.131i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.974364517\)
\(L(\frac12)\) \(\approx\) \(1.974364517\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.28 - 1.33i)T \)
good5 \( 1 + (-0.779 + 1.35i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.16iT - 13T^{2} \)
17 \( 1 + (-1.33 - 2.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.08 - 1.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.21 - 4.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.56iT - 29T^{2} \)
31 \( 1 + (-7.47 + 4.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.31 + 5.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.84T + 41T^{2} \)
43 \( 1 + 6.78T + 43T^{2} \)
47 \( 1 + (-2.84 + 4.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.92 - 3.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.61 + 7.99i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.9 - 6.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.13 - 5.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-2.65 + 1.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.61 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.47T + 83T^{2} \)
89 \( 1 + (-1.96 + 3.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.46iT - 97T^{2} \)
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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

−9.547234576940018185793278569976, −8.497528652957703494436021680665, −7.933834671450527625843940932038, −7.31965678737258277838790834806, −5.82952276966714312608880329846, −5.30668040916602493074670779875, −4.74466286364026738160814465898, −3.28452562972614614875536814215, −2.27114400691613573005071229127, −1.05202072642102575827894789252, 1.06220907524795126710576117942, 2.48062821430638286682587025769, 3.25742061108292507399703941425, 4.77981657965508445723619149952, 5.04217549277962886000911624968, 6.48680014971725133355003467114, 6.94157047512459564775953074583, 7.933561962959954885574302308652, 8.600958050635143746105241347853, 9.540073835931203790187984276663

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