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Properties

Label 2-1512-63.4-c1-0-18
Degree $2$
Conductor $1512$
Sign $-0.840 + 0.541i$
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  − 2.66·5-s + (1.94 + 1.79i)7-s − 1.36·11-s + (−2.75 + 4.77i)13-s + (1.23 − 2.14i)17-s + (−2.19 − 3.80i)19-s + 4.69·23-s + 2.11·25-s + (−2.94 − 5.10i)29-s + (−1.55 − 2.69i)31-s + (−5.18 − 4.78i)35-s + (−3.15 − 5.46i)37-s + (−1.38 + 2.40i)41-s + (−4.87 − 8.45i)43-s + (−5.02 + 8.70i)47-s + ⋯
L(s)  = 1  − 1.19·5-s + (0.735 + 0.678i)7-s − 0.411·11-s + (−0.764 + 1.32i)13-s + (0.300 − 0.520i)17-s + (−0.503 − 0.872i)19-s + 0.977·23-s + 0.423·25-s + (−0.547 − 0.948i)29-s + (−0.279 − 0.484i)31-s + (−0.877 − 0.809i)35-s + (−0.518 − 0.898i)37-s + (−0.216 + 0.375i)41-s + (−0.744 − 1.28i)43-s + (−0.732 + 1.26i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.840 + 0.541i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1663179329\)
\(L(\frac12)\) \(\approx\) \(0.1663179329\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.79i)T \)
good5 \( 1 + 2.66T + 5T^{2} \)
11 \( 1 + 1.36T + 11T^{2} \)
13 \( 1 + (2.75 - 4.77i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.23 + 2.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.19 + 3.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.69T + 23T^{2} \)
29 \( 1 + (2.94 + 5.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.55 + 2.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.15 + 5.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.38 - 2.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.87 + 8.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.02 - 8.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.47 + 2.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.77 - 3.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.663 - 1.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.11 - 1.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.41 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.15 + 8.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.73 + 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.55 + 4.42i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.004918969285443742074452748047, −8.393396259336306816234859862510, −7.43745118305897653246949530794, −7.05503156908089479955497177336, −5.74577937167364561544973676041, −4.76708467502819274014874866645, −4.25141706297629270104874239398, −2.96418006845078709542505138863, −1.93467869081988478542573392173, −0.06631506867289293078747071239, 1.42756606069867503428057265739, 3.05256532852978542070694026306, 3.80469422245992172257469059175, 4.81865637416757531439239664267, 5.45380734672867566638664054713, 6.83016875775210705068342843999, 7.58477360237240029217540752300, 8.062370746094718691954932761380, 8.698417800268654762278424062955, 10.12295141089668385570953124892

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