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Properties

Label 2-1512-63.4-c1-0-4
Degree $2$
Conductor $1512$
Sign $0.657 - 0.753i$
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  − 3.40·5-s + (−2.05 − 1.66i)7-s − 5.39·11-s + (1.89 − 3.28i)13-s + (−0.411 + 0.713i)17-s + (0.233 + 0.404i)19-s + 5.49·23-s + 6.61·25-s + (−0.400 − 0.693i)29-s + (4.95 + 8.57i)31-s + (7.01 + 5.66i)35-s + (4.34 + 7.52i)37-s + (−1.84 + 3.19i)41-s + (−4.36 − 7.55i)43-s + (−5.24 + 9.09i)47-s + ⋯
L(s)  = 1  − 1.52·5-s + (−0.778 − 0.628i)7-s − 1.62·11-s + (0.525 − 0.910i)13-s + (−0.0999 + 0.173i)17-s + (0.0535 + 0.0928i)19-s + 1.14·23-s + 1.32·25-s + (−0.0743 − 0.128i)29-s + (0.889 + 1.54i)31-s + (1.18 + 0.957i)35-s + (0.713 + 1.23i)37-s + (−0.288 + 0.498i)41-s + (−0.665 − 1.15i)43-s + (−0.765 + 1.32i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6415563553\)
\(L(\frac12)\) \(\approx\) \(0.6415563553\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 1.66i)T \)
good5 \( 1 + 3.40T + 5T^{2} \)
11 \( 1 + 5.39T + 11T^{2} \)
13 \( 1 + (-1.89 + 3.28i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.411 - 0.713i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.233 - 0.404i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.49T + 23T^{2} \)
29 \( 1 + (0.400 + 0.693i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.95 - 8.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.34 - 7.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.84 - 3.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.36 + 7.55i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.24 - 9.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.71 + 8.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.830 + 1.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.474 - 0.821i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.269 + 0.466i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 + (-2.58 + 4.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.91 - 6.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.79 - 6.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.73 - 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.22 + 5.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.721629567902612784619688117494, −8.361753800243502094442764084926, −8.128394562391683509332105240812, −7.25291605645107438749214144714, −6.54777385537380628312638147625, −5.28837725567650340171245891838, −4.52833354211233365714395125500, −3.36769114102830078102125022093, −2.98112129616841020002464132646, −0.802969623268974375859778201037, 0.37325975223439459211811038836, 2.43940457643278626194240245766, 3.29434635441108749199716182750, 4.23695712236003348193295496955, 5.12564321696582891285913171624, 6.14613722488130104105655444521, 7.10080061317640756350448430755, 7.76792705120983626713664767478, 8.512979569826392626403113784945, 9.231688078608702284523778485022

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