Properties

Label 2-504-56.37-c1-0-33
Degree $2$
Conductor $504$
Sign $0.715 + 0.698i$
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  + (1.40 + 0.114i)2-s + (1.97 + 0.323i)4-s + (−2.80 − 1.61i)5-s + (1.47 − 2.19i)7-s + (2.74 + 0.682i)8-s + (−3.76 − 2.60i)10-s + (2.08 − 1.20i)11-s − 3.09i·13-s + (2.33 − 2.92i)14-s + (3.79 + 1.27i)16-s + (1.97 + 3.42i)17-s + (−2.33 − 1.35i)19-s + (−5.01 − 4.10i)20-s + (3.08 − 1.46i)22-s + (1.37 − 2.37i)23-s + ⋯
L(s)  = 1  + (0.996 + 0.0811i)2-s + (0.986 + 0.161i)4-s + (−1.25 − 0.724i)5-s + (0.558 − 0.829i)7-s + (0.970 + 0.241i)8-s + (−1.19 − 0.823i)10-s + (0.629 − 0.363i)11-s − 0.858i·13-s + (0.624 − 0.781i)14-s + (0.947 + 0.319i)16-s + (0.479 + 0.830i)17-s + (−0.536 − 0.309i)19-s + (−1.12 − 0.917i)20-s + (0.657 − 0.311i)22-s + (0.286 − 0.495i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.715 + 0.698i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.715 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18648 - 0.889836i\)
\(L(\frac12)\) \(\approx\) \(2.18648 - 0.889836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.114i)T \)
3 \( 1 \)
7 \( 1 + (-1.47 + 2.19i)T \)
good5 \( 1 + (2.80 + 1.61i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + (-1.97 - 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.33 + 1.35i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.37 + 2.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (-1.10 - 1.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.30 + 2.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.11T + 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 + (3.31 - 5.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.23 + 1.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.6 - 6.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.54 - 4.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.01 - 2.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.64T + 71T^{2} \)
73 \( 1 + (-4.77 - 8.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.838 - 1.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.47iT - 83T^{2} \)
89 \( 1 + (-6.98 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.37T + 97T^{2} \)
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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

−11.05337599237799803660243530518, −10.33826117532584257571592548959, −8.643023560532791468833288330080, −7.971408210500791880590006029912, −7.21112616671264106476601298722, −6.04883614490989887244570814195, −4.81049623427925260864298851410, −4.16386828165613137477804949707, −3.24744209275960031182481738767, −1.18581613544742625060607029004, 2.01979736031668936670395038712, 3.32411362174430300393932118734, 4.21981076698701421628780856933, 5.20653139922094298250695677292, 6.45704601107367414824733238752, 7.22706895452245931971092936639, 8.055416897425837894277491108673, 9.261597276396987362849294347151, 10.51841509362379697135255577248, 11.46778604388353310883749953058

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