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Properties

Label 2-1512-63.25-c1-0-11
Degree $2$
Conductor $1512$
Sign $0.999 + 0.0286i$
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  + (−0.234 + 0.405i)5-s + (0.212 + 2.63i)7-s + (−0.674 − 1.16i)11-s + (−3.16 − 5.48i)13-s + (2.47 − 4.28i)17-s + (2.38 + 4.13i)19-s + (3.81 − 6.60i)23-s + (2.39 + 4.14i)25-s + (1.80 − 3.12i)29-s + 6.49·31-s + (−1.11 − 0.531i)35-s + (5.24 + 9.07i)37-s + (0.0251 + 0.0435i)41-s + (−0.431 + 0.748i)43-s + 10.9·47-s + ⋯
L(s)  = 1  + (−0.104 + 0.181i)5-s + (0.0802 + 0.996i)7-s + (−0.203 − 0.352i)11-s + (−0.877 − 1.52i)13-s + (0.599 − 1.03i)17-s + (0.548 + 0.949i)19-s + (0.795 − 1.37i)23-s + (0.478 + 0.828i)25-s + (0.335 − 0.580i)29-s + 1.16·31-s + (−0.189 − 0.0897i)35-s + (0.861 + 1.49i)37-s + (0.00392 + 0.00680i)41-s + (−0.0658 + 0.114i)43-s + 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.0286i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.999 + 0.0286i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642993783\)
\(L(\frac12)\) \(\approx\) \(1.642993783\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.212 - 2.63i)T \)
good5 \( 1 + (0.234 - 0.405i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.674 + 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.16 + 5.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.47 + 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 4.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.81 + 6.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.80 + 3.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.49T + 31T^{2} \)
37 \( 1 + (-5.24 - 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0251 - 0.0435i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.431 - 0.748i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + (5.84 - 10.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.87T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 + 2.64T + 67T^{2} \)
71 \( 1 - 7.04T + 71T^{2} \)
73 \( 1 + (3.30 - 5.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.17T + 79T^{2} \)
83 \( 1 + (4.90 - 8.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.30 + 9.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.97 + 12.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.597107989403680905251183418897, −8.528565247906863647736525827344, −7.966852938638307324695326216612, −7.16034264961270942612698164948, −6.04506400747812459081357026604, −5.37275634161975258415786847108, −4.62922422236965574666829191070, −2.92656361092817244252088201255, −2.80637933163176734621905650232, −0.879175338181499437270860572369, 0.992984205668851481610262379731, 2.27537998590002060017959581601, 3.58753282004838115964107946707, 4.46774651216065234781333841583, 5.12080464483942383163351108330, 6.39990032847290569934090423831, 7.17663374505600432747881735042, 7.67129505641910716033893254076, 8.764423432614814619478667166945, 9.521204466726544826953376441947

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