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Properties

Label 2-1512-24.11-c1-0-57
Degree $2$
Conductor $1512$
Sign $0.707 - 0.707i$
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + 2.38·5-s − i·7-s + (−2 − 1.99i)8-s + (0.873 + 3.25i)10-s − 1.65i·11-s − 0.253i·13-s + (1.36 − 0.366i)14-s + (1.99 − 3.46i)16-s − 2i·17-s + 7.03·19-s + (−4.13 + 2.38i)20-s + (2.25 − 0.605i)22-s + 2.82·23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + 1.06·5-s − 0.377i·7-s + (−0.707 − 0.707i)8-s + (0.276 + 1.03i)10-s − 0.498i·11-s − 0.0702i·13-s + (0.365 − 0.0978i)14-s + (0.499 − 0.866i)16-s − 0.485i·17-s + 1.61·19-s + (−0.924 + 0.533i)20-s + (0.481 − 0.129i)22-s + 0.589·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166145717\)
\(L(\frac12)\) \(\approx\) \(2.166145717\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 2.38T + 5T^{2} \)
11 \( 1 + 1.65iT - 11T^{2} \)
13 \( 1 + 0.253iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 7.03T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 1.26T + 29T^{2} \)
31 \( 1 + 0.746iT - 31T^{2} \)
37 \( 1 + 6.94iT - 37T^{2} \)
41 \( 1 - 5.55iT - 41T^{2} \)
43 \( 1 - 8.67T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 3.30T + 53T^{2} \)
59 \( 1 - 10.0iT - 59T^{2} \)
61 \( 1 + 4.53iT - 61T^{2} \)
67 \( 1 - 0.253T + 67T^{2} \)
71 \( 1 - 3.17T + 71T^{2} \)
73 \( 1 + 3.05T + 73T^{2} \)
79 \( 1 + 14.5iT - 79T^{2} \)
83 \( 1 - 1.77iT - 83T^{2} \)
89 \( 1 + 3.13iT - 89T^{2} \)
97 \( 1 + 5.49T + 97T^{2} \)
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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

−9.363423027115613028062243393102, −8.893118162675505692426748636737, −7.65772327760946563548669330291, −7.25294152696638594483666298300, −6.13474083203964939614265164145, −5.63427216463293157716647156875, −4.83677028915946442819404553980, −3.70501310319092415716534079714, −2.68423024587464121861860401845, −0.959285290487153004088968285354, 1.22403549355326587184619081755, 2.20554456209787658899865059558, 3.09460878078627720020357216256, 4.22445760203080088821925741166, 5.31807865387279948842418844078, 5.70534088024178702330153865360, 6.80474129635942831167024837632, 7.935241168963312145554111139271, 9.014060740404631814703726637913, 9.489847739235086544939832213600

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