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Properties

Label 2-1512-63.47-c1-0-14
Degree $2$
Conductor $1512$
Sign $-0.0761 + 0.997i$
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  − 2.76·5-s + (−1.21 + 2.34i)7-s − 2.41i·11-s + (0.0461 − 0.0266i)13-s + (1.79 + 3.11i)17-s + (3.96 + 2.29i)19-s + 0.106i·23-s + 2.66·25-s + (−6.57 − 3.79i)29-s + (−5.90 − 3.40i)31-s + (3.37 − 6.50i)35-s + (2.06 − 3.57i)37-s + (−0.838 − 1.45i)41-s + (1.74 − 3.01i)43-s + (−6.59 − 11.4i)47-s + ⋯
L(s)  = 1  − 1.23·5-s + (−0.460 + 0.887i)7-s − 0.729i·11-s + (0.0128 − 0.00739i)13-s + (0.435 + 0.755i)17-s + (0.910 + 0.525i)19-s + 0.0222i·23-s + 0.533·25-s + (−1.22 − 0.704i)29-s + (−1.06 − 0.612i)31-s + (0.570 − 1.09i)35-s + (0.338 − 0.587i)37-s + (−0.131 − 0.226i)41-s + (0.265 − 0.460i)43-s + (−0.962 − 1.66i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0761 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0761 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.0761 + 0.997i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.0761 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6410876073\)
\(L(\frac12)\) \(\approx\) \(0.6410876073\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 - 2.34i)T \)
good5 \( 1 + 2.76T + 5T^{2} \)
11 \( 1 + 2.41iT - 11T^{2} \)
13 \( 1 + (-0.0461 + 0.0266i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.79 - 3.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.96 - 2.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.106iT - 23T^{2} \)
29 \( 1 + (6.57 + 3.79i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.90 + 3.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.06 + 3.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.838 + 1.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.74 + 3.01i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.59 + 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.27 + 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.72 + 8.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.29 + 1.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.09 + 12.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.40iT - 71T^{2} \)
73 \( 1 + (-5.54 + 3.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.11 + 3.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.86 - 11.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.71 - 9.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.3 - 8.28i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.230643970210821105942552102823, −8.302172767047673535729778351595, −7.85889673308664356860358488851, −6.91701721764262488256260116389, −5.84983459717147114564372236901, −5.29399046204204928759282208290, −3.73281422951865150630516081108, −3.55264204678051271293612187547, −2.08722962318531969016690705048, −0.29084566140185731343743116243, 1.14049144005668259876230130695, 2.91905998025191179941779085897, 3.73276998159757379058099942077, 4.51555961710508037116997295354, 5.42925771097217107809799274889, 6.77287681101987432066582446981, 7.39813456940976600140456611920, 7.75306649255509746502768789995, 8.960247128625690448482467058140, 9.668437387655281287374228792996

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