Properties

Label 2-504-63.47-c1-0-23
Degree $2$
Conductor $504$
Sign $-0.995 - 0.0948i$
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  + (−0.601 − 1.62i)3-s − 0.207·5-s + (−1.37 − 2.26i)7-s + (−2.27 + 1.95i)9-s − 1.51i·11-s + (−3.39 + 1.95i)13-s + (0.124 + 0.336i)15-s + (−0.873 − 1.51i)17-s + (−0.968 − 0.559i)19-s + (−2.85 + 3.58i)21-s + 1.44i·23-s − 4.95·25-s + (4.54 + 2.52i)27-s + (1.99 + 1.15i)29-s + (−5.15 − 2.97i)31-s + ⋯
L(s)  = 1  + (−0.347 − 0.937i)3-s − 0.0925·5-s + (−0.518 − 0.855i)7-s + (−0.759 + 0.651i)9-s − 0.457i·11-s + (−0.940 + 0.543i)13-s + (0.0321 + 0.0868i)15-s + (−0.211 − 0.366i)17-s + (−0.222 − 0.128i)19-s + (−0.622 + 0.782i)21-s + 0.301i·23-s − 0.991·25-s + (0.873 + 0.485i)27-s + (0.370 + 0.214i)29-s + (−0.926 − 0.534i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.995 - 0.0948i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.995 - 0.0948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0223363 + 0.470154i\)
\(L(\frac12)\) \(\approx\) \(0.0223363 + 0.470154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.601 + 1.62i)T \)
7 \( 1 + (1.37 + 2.26i)T \)
good5 \( 1 + 0.207T + 5T^{2} \)
11 \( 1 + 1.51iT - 11T^{2} \)
13 \( 1 + (3.39 - 1.95i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.873 + 1.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.968 + 0.559i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.44iT - 23T^{2} \)
29 \( 1 + (-1.99 - 1.15i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.15 + 2.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.13 - 3.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.91 + 5.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.213 - 0.369i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.13 + 7.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.16 + 4.71i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.63 + 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.24 + 5.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.10iT - 71T^{2} \)
73 \( 1 + (-10.6 + 6.13i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.57 + 4.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.36 + 14.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.96 + 3.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.1 + 7.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.55945219356715140989326658124, −9.638684516156021496651629680514, −8.546073429498125079782320973946, −7.45878315060275815205069872304, −6.95004383002640489671002905897, −5.96591629670206158336874335243, −4.81463256696644527984079985672, −3.46033574191847758495398468825, −2.01400516564623472298745799943, −0.27365169488911626955831402398, 2.45326713541236596590609789738, 3.65583158291698981930638432839, 4.82214059533911199644742175089, 5.66876521714010343261510619289, 6.60204719166888569022291774068, 7.902957572123216023765986415873, 8.943680165095461282911191399803, 9.684831989877218990743705626924, 10.35357987245725817289324491732, 11.31249738703780868073218371325

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