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Properties

Label 2-504-21.5-c3-0-23
Degree $2$
Conductor $504$
Sign $-0.977 + 0.212i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.74 − 16.8i)5-s + (−18.4 − 1.67i)7-s + (15.1 − 8.72i)11-s + 16.5i·13-s + (−68.6 − 118. i)17-s + (−20.6 − 11.9i)19-s + (108. + 62.8i)23-s + (−127. − 220. i)25-s + 105. i·29-s + (−95.0 + 54.8i)31-s + (−207. + 294. i)35-s + (58.5 − 101. i)37-s − 348.·41-s − 141.·43-s + (−172. + 299. i)47-s + ⋯
L(s)  = 1  + (0.871 − 1.50i)5-s + (−0.995 − 0.0902i)7-s + (0.414 − 0.239i)11-s + 0.352i·13-s + (−0.979 − 1.69i)17-s + (−0.249 − 0.144i)19-s + (0.987 + 0.570i)23-s + (−1.01 − 1.76i)25-s + 0.673i·29-s + (−0.550 + 0.318i)31-s + (−1.00 + 1.42i)35-s + (0.260 − 0.450i)37-s − 1.32·41-s − 0.503·43-s + (−0.536 + 0.929i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.977 + 0.212i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -0.977 + 0.212i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.095010258\)
\(L(\frac12)\) \(\approx\) \(1.095010258\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 + 1.67i)T \)
good5 \( 1 + (-9.74 + 16.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-15.1 + 8.72i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 16.5iT - 2.19e3T^{2} \)
17 \( 1 + (68.6 + 118. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (20.6 + 11.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-108. - 62.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 105. iT - 2.43e4T^{2} \)
31 \( 1 + (95.0 - 54.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-58.5 + 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 348.T + 6.89e4T^{2} \)
43 \( 1 + 141.T + 7.95e4T^{2} \)
47 \( 1 + (172. - 299. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (149. - 86.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-297. - 515. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (561. + 324. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (64.9 + 112. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 908. iT - 3.57e5T^{2} \)
73 \( 1 + (44.8 - 25.9i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-474. + 822. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.30e3T + 5.71e5T^{2} \)
89 \( 1 + (-210. + 364. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 553. iT - 9.12e5T^{2} \)
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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

−9.730797347768373263703464530944, −9.188195077067553053554930215412, −8.749457750140090244266054052180, −7.20787652251374410312536406981, −6.34780464382242724144713944732, −5.27848701456186269459807469610, −4.52246804394514529601072498454, −3.04612509433918388069908818594, −1.58708446004495217342751418697, −0.31947292206998930613121013872, 1.92011127612185081767117765435, 2.94889194237432404036394349866, 3.93886503558520955044972816781, 5.62550961328083780157083922908, 6.62537057371350629957213953294, 6.74591848755115654488550362482, 8.281986471179222164513213440134, 9.357418813855508003544264389689, 10.19292477728473534503187119930, 10.64884638349493946121840831017

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