Properties

Label 2-1512-24.11-c1-0-13
Degree $2$
Conductor $1512$
Sign $-0.340 - 0.940i$
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  + (−1.29 − 0.560i)2-s + (1.37 + 1.45i)4-s + 0.530·5-s + i·7-s + (−0.962 − 2.65i)8-s + (−0.688 − 0.297i)10-s + 0.905i·11-s + 4.38i·13-s + (0.560 − 1.29i)14-s + (−0.242 + 3.99i)16-s − 1.25i·17-s + 0.00364·19-s + (0.726 + 0.772i)20-s + (0.507 − 1.17i)22-s − 0.778·23-s + ⋯
L(s)  = 1  + (−0.917 − 0.396i)2-s + (0.685 + 0.728i)4-s + 0.237·5-s + 0.377i·7-s + (−0.340 − 0.940i)8-s + (−0.217 − 0.0940i)10-s + 0.272i·11-s + 1.21i·13-s + (0.149 − 0.346i)14-s + (−0.0606 + 0.998i)16-s − 0.304i·17-s + 0.000837·19-s + (0.162 + 0.172i)20-s + (0.108 − 0.250i)22-s − 0.162·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.340 - 0.940i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.340 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6353059211\)
\(L(\frac12)\) \(\approx\) \(0.6353059211\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.560i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.530T + 5T^{2} \)
11 \( 1 - 0.905iT - 11T^{2} \)
13 \( 1 - 4.38iT - 13T^{2} \)
17 \( 1 + 1.25iT - 17T^{2} \)
19 \( 1 - 0.00364T + 19T^{2} \)
23 \( 1 + 0.778T + 23T^{2} \)
29 \( 1 + 5.43T + 29T^{2} \)
31 \( 1 - 3.49iT - 31T^{2} \)
37 \( 1 - 5.34iT - 37T^{2} \)
41 \( 1 - 1.74iT - 41T^{2} \)
43 \( 1 + 0.259T + 43T^{2} \)
47 \( 1 - 3.75T + 47T^{2} \)
53 \( 1 + 3.08T + 53T^{2} \)
59 \( 1 + 5.73iT - 59T^{2} \)
61 \( 1 + 0.555iT - 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 + 1.96iT - 79T^{2} \)
83 \( 1 - 5.82iT - 83T^{2} \)
89 \( 1 + 1.94iT - 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−9.606228326092917670991315605919, −9.089608881065666506873699865719, −8.287186088439647637068633339139, −7.40247944985942318562341353700, −6.68398819655319173917367507053, −5.81921000050804158577505183209, −4.56514127908919737513661730122, −3.53710117331141562097039816668, −2.37250862538486730324953691529, −1.53166347162409278473534818503, 0.33078582165476791461057051719, 1.70169193226522898372859746813, 2.89421235906687930552707910062, 4.12608893028548586066673303649, 5.58134334196458224616633649182, 5.85274379714290864224599839089, 7.03461248649009847357498405638, 7.70966438789585044142227475487, 8.349415969491877186628692462946, 9.250682082490267592437496417459

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