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Properties

Label 2-1512-24.11-c1-0-5
Degree $2$
Conductor $1512$
Sign $-0.893 + 0.449i$
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  + (−0.508 + 1.31i)2-s + (−1.48 − 1.34i)4-s − 2.29·5-s − i·7-s + (2.52 − 1.27i)8-s + (1.16 − 3.03i)10-s + 4.02i·11-s − 1.82i·13-s + (1.31 + 0.508i)14-s + (0.393 + 3.98i)16-s + 0.430i·17-s + 5.01·19-s + (3.40 + 3.08i)20-s + (−5.30 − 2.04i)22-s + 3.49·23-s + ⋯
L(s)  = 1  + (−0.359 + 0.933i)2-s + (−0.741 − 0.671i)4-s − 1.02·5-s − 0.377i·7-s + (0.893 − 0.449i)8-s + (0.369 − 0.958i)10-s + 1.21i·11-s − 0.506i·13-s + (0.352 + 0.136i)14-s + (0.0982 + 0.995i)16-s + 0.104i·17-s + 1.14·19-s + (0.761 + 0.689i)20-s + (−1.13 − 0.436i)22-s + 0.728·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.893 + 0.449i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.893 + 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2510584121\)
\(L(\frac12)\) \(\approx\) \(0.2510584121\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.508 - 1.31i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 2.29T + 5T^{2} \)
11 \( 1 - 4.02iT - 11T^{2} \)
13 \( 1 + 1.82iT - 13T^{2} \)
17 \( 1 - 0.430iT - 17T^{2} \)
19 \( 1 - 5.01T + 19T^{2} \)
23 \( 1 - 3.49T + 23T^{2} \)
29 \( 1 + 2.16T + 29T^{2} \)
31 \( 1 + 2.10iT - 31T^{2} \)
37 \( 1 + 2.19iT - 37T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 1.72T + 47T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 - 8.44iT - 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 + 7.16T + 73T^{2} \)
79 \( 1 + 15.2iT - 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 + 7.44iT - 89T^{2} \)
97 \( 1 - 1.68T + 97T^{2} \)
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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

−9.817703134717784153331640158517, −9.030240786503032320569277552844, −8.080575426231219366910600576836, −7.45904648559430281624166073569, −7.07824590109256688351502630917, −5.94407429629899546266379412670, −4.93082990814356755375432171757, −4.29656432712290827494367015372, −3.23564704274264051431244939571, −1.38847420706924228112079064989, 0.12481655924085978392447974915, 1.51947473014610619561532846863, 3.05935805849004495537618690597, 3.49612739144872000598478121185, 4.61081485765494569456066682117, 5.47425387203489270218024287701, 6.77393717791363088236196721898, 7.73105491814355762544991248891, 8.338530067449021611656883514778, 9.043562073300518489140085176486

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