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Properties

Label 2-504-168.125-c1-0-15
Degree $2$
Conductor $504$
Sign $0.311 + 0.950i$
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  + (−0.581 − 1.28i)2-s + (−1.32 + 1.50i)4-s − 2.64·7-s + (2.70 + 0.832i)8-s + 6.57·11-s + (1.53 + 3.41i)14-s + (−0.5 − 3.96i)16-s + (−3.82 − 8.46i)22-s − 9.39i·23-s + 5·25-s + (3.50 − 3.96i)28-s + 8.89·29-s + (−4.82 + 2.95i)32-s + 6i·37-s − 12i·43-s + (−8.69 + 9.85i)44-s + ⋯
L(s)  = 1  + (−0.411 − 0.911i)2-s + (−0.661 + 0.750i)4-s − 0.999·7-s + (0.955 + 0.294i)8-s + 1.98·11-s + (0.411 + 0.911i)14-s + (−0.125 − 0.992i)16-s + (−0.815 − 1.80i)22-s − 1.95i·23-s + 25-s + (0.661 − 0.749i)28-s + 1.65·29-s + (−0.852 + 0.522i)32-s + 0.986i·37-s − 1.82i·43-s + (−1.31 + 1.48i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870004 - 0.630357i\)
\(L(\frac12)\) \(\approx\) \(0.870004 - 0.630357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.581 + 1.28i)T \)
3 \( 1 \)
7 \( 1 + 2.64T \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 6.57T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 9.39iT - 23T^{2} \)
29 \( 1 - 8.89T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 0.412T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 7.57iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.55694232071315057406852396082, −9.996646463366611420098839319947, −8.930004839889490449999813484436, −8.600232726908091979219000242971, −6.97848949740831147291397864361, −6.38210237283518228559044639221, −4.63968673812115239321567389342, −3.72715818049701084708443148730, −2.63688905819385312588039680444, −0.962277208072123802136973814267, 1.22163493667895208101562606174, 3.41910138215955781332084927030, 4.50442903716713647660058357624, 5.86389487958786794346716686194, 6.57754293100539889100189171135, 7.26419286559349298738568450251, 8.507061232063857410822346221142, 9.364227779999202578665029139483, 9.739653379398933416230047003935, 10.95238687560792854793104525893

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