Properties

Label 2-504-21.20-c1-0-6
Degree $2$
Conductor $504$
Sign $0.981 + 0.192i$
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  + 3.69·5-s + (2.41 − 1.08i)7-s − 3.41i·11-s + 5.22i·13-s − 6.75·17-s − 2.16i·19-s − 6.24i·23-s + 8.65·25-s + 2.58i·29-s + 10.4i·31-s + (8.92 − 4i)35-s + 4·37-s + 0.634·41-s − 6.48·43-s + 3.06·47-s + ⋯
L(s)  = 1  + 1.65·5-s + (0.912 − 0.409i)7-s − 1.02i·11-s + 1.44i·13-s − 1.63·17-s − 0.496i·19-s − 1.30i·23-s + 1.73·25-s + 0.480i·29-s + 1.87i·31-s + (1.50 − 0.676i)35-s + 0.657·37-s + 0.0990·41-s − 0.988·43-s + 0.446·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.981 + 0.192i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.981 + 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95423 - 0.190163i\)
\(L(\frac12)\) \(\approx\) \(1.95423 - 0.190163i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 1.08i)T \)
good5 \( 1 - 3.69T + 5T^{2} \)
11 \( 1 + 3.41iT - 11T^{2} \)
13 \( 1 - 5.22iT - 13T^{2} \)
17 \( 1 + 6.75T + 17T^{2} \)
19 \( 1 + 2.16iT - 19T^{2} \)
23 \( 1 + 6.24iT - 23T^{2} \)
29 \( 1 - 2.58iT - 29T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 0.634T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + 2.58iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 - 7.41iT - 71T^{2} \)
73 \( 1 + 0.896iT - 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 6.75T + 89T^{2} \)
97 \( 1 - 9.55iT - 97T^{2} \)
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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

−10.89196802669013637717380533377, −10.08648367428099781199782430889, −8.824571653571577119899960262381, −8.719165500165490722006384332651, −6.90097909596582414979576552196, −6.41521832068426138337057421110, −5.20414072250476346171854627163, −4.36423386406142296763981524472, −2.55925388331351432594142459544, −1.51818637491275169410531133793, 1.72403811441065086704291898510, 2.54623487828967201266365030209, 4.43930551243073396443190389015, 5.47060514494886720419163836010, 6.05769009748373983838584301429, 7.34253536139718242448809513035, 8.309850792244481214099251876119, 9.382512751085801744499345045735, 9.935705862463291596244157975233, 10.84183575931821259982259126577

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