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Properties

Label 2-504-24.11-c1-0-14
Degree $2$
Conductor $504$
Sign $-0.277 + 0.960i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−0.157 − 1.40i)2-s + (−1.95 + 0.442i)4-s + 1.81·5-s − i·7-s + (0.928 + 2.67i)8-s + (−0.286 − 2.55i)10-s + 2.36i·11-s − 6.99i·13-s + (−1.40 + 0.157i)14-s + (3.60 − 1.72i)16-s − 3.69i·17-s + 5.53·19-s + (−3.54 + 0.804i)20-s + (3.32 − 0.372i)22-s + 7.60·23-s + ⋯
L(s)  = 1  + (−0.111 − 0.993i)2-s + (−0.975 + 0.221i)4-s + 0.813·5-s − 0.377i·7-s + (0.328 + 0.944i)8-s + (−0.0904 − 0.808i)10-s + 0.714i·11-s − 1.93i·13-s + (−0.375 + 0.0420i)14-s + (0.902 − 0.431i)16-s − 0.895i·17-s + 1.26·19-s + (−0.793 + 0.179i)20-s + (0.709 − 0.0794i)22-s + 1.58·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.277 + 0.960i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.830358 - 1.10411i\)
\(L(\frac12)\) \(\approx\) \(0.830358 - 1.10411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.157 + 1.40i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 1.81T + 5T^{2} \)
11 \( 1 - 2.36iT - 11T^{2} \)
13 \( 1 + 6.99iT - 13T^{2} \)
17 \( 1 + 3.69iT - 17T^{2} \)
19 \( 1 - 5.53T + 19T^{2} \)
23 \( 1 - 7.60T + 23T^{2} \)
29 \( 1 + 5.87T + 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 + 8.88iT - 37T^{2} \)
41 \( 1 - 9.83iT - 41T^{2} \)
43 \( 1 + 2.88T + 43T^{2} \)
47 \( 1 + 0.169T + 47T^{2} \)
53 \( 1 - 2.02T + 53T^{2} \)
59 \( 1 + 0.169iT - 59T^{2} \)
61 \( 1 + 0.420iT - 61T^{2} \)
67 \( 1 + 9.76T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 8.78T + 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 - 7.42iT - 89T^{2} \)
97 \( 1 - 4.84T + 97T^{2} \)
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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

−10.66594463558369572640373096559, −9.675981182911831932625948659441, −9.440766484125997515877866778862, −8.014632385515702256844556736912, −7.25226876458772736469136699327, −5.57939450703873243862207479028, −5.00219403620186067603239619947, −3.48999710719127701904059331114, −2.51168009584855176946493292481, −0.976576863281343686977864083987, 1.62863123365652294256155580193, 3.52048962210387616782519040072, 4.85268488168025752480456588182, 5.74869772823679651585476577266, 6.54345579365101396320059392328, 7.39671729825711573968686168971, 8.700236143876635921635658717281, 9.154468494302831110676185957225, 9.956983517349554934759036553144, 11.11530552262152150942312791944

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