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Properties

Label 2-1512-63.16-c1-0-13
Degree $2$
Conductor $1512$
Sign $-0.117 + 0.993i$
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  − 1.85·5-s + (−2.60 + 0.464i)7-s + 2.57·11-s + (2.82 + 4.88i)13-s + (−3.57 − 6.19i)17-s + (0.636 − 1.10i)19-s − 0.241·23-s − 1.55·25-s + (−0.923 + 1.59i)29-s + (1.49 − 2.59i)31-s + (4.83 − 0.862i)35-s + (0.338 − 0.585i)37-s + (0.733 + 1.27i)41-s + (4.14 − 7.17i)43-s + (−6.15 − 10.6i)47-s + ⋯
L(s)  = 1  − 0.829·5-s + (−0.984 + 0.175i)7-s + 0.776·11-s + (0.782 + 1.35i)13-s + (−0.868 − 1.50i)17-s + (0.146 − 0.252i)19-s − 0.0503·23-s − 0.311·25-s + (−0.171 + 0.297i)29-s + (0.268 − 0.465i)31-s + (0.817 − 0.145i)35-s + (0.0556 − 0.0963i)37-s + (0.114 + 0.198i)41-s + (0.631 − 1.09i)43-s + (−0.898 − 1.55i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7606516495\)
\(L(\frac12)\) \(\approx\) \(0.7606516495\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 0.464i)T \)
good5 \( 1 + 1.85T + 5T^{2} \)
11 \( 1 - 2.57T + 11T^{2} \)
13 \( 1 + (-2.82 - 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.57 + 6.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.636 + 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.241T + 23T^{2} \)
29 \( 1 + (0.923 - 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.49 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.338 + 0.585i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.733 - 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.14 + 7.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.15 + 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.35 + 5.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.04 + 1.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.47 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.41 + 4.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.53T + 71T^{2} \)
73 \( 1 + (6.55 + 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.86 - 3.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.00 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.60 - 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.40 + 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.188280407783604217242024439051, −8.694035367764914981254092825042, −7.51339638708299078537181358707, −6.76556177026999285900210288258, −6.27972232019827393096289118331, −4.95162404864806685765333455122, −4.02431624222449896565278897434, −3.36560550117415477587191766928, −2.05656521040970241399811689754, −0.33366248581158080777507961792, 1.20687208828013440672126790149, 2.91549669254090143833775209670, 3.75272682637012065054052769389, 4.35791892885945292531926555922, 5.92300327234851697551635526202, 6.26042346472699244662375087311, 7.35718558245108458798310392428, 8.122182415799809773661077887984, 8.786259302494264801874087529647, 9.704347453358243665414299366566

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