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Properties

Label 2-504-21.5-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.952 + 0.305i$
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  + (−1.45 + 2.51i)5-s + (−2.64 + 0.0146i)7-s + (−1.08 + 0.628i)11-s − 5.32i·13-s + (−0.880 − 1.52i)17-s + (−6.69 − 3.86i)19-s + (−4.43 − 2.55i)23-s + (−1.72 − 2.98i)25-s + 8.74i·29-s + (2.18 − 1.26i)31-s + (3.80 − 6.68i)35-s + (−3.66 + 6.34i)37-s + 3.09·41-s + 1.15·43-s + (−3.44 + 5.96i)47-s + ⋯
L(s)  = 1  + (−0.649 + 1.12i)5-s + (−0.999 + 0.00554i)7-s + (−0.328 + 0.189i)11-s − 1.47i·13-s + (−0.213 − 0.369i)17-s + (−1.53 − 0.887i)19-s + (−0.924 − 0.533i)23-s + (−0.344 − 0.597i)25-s + 1.62i·29-s + (0.392 − 0.226i)31-s + (0.643 − 1.12i)35-s + (−0.602 + 1.04i)37-s + 0.483·41-s + 0.176·43-s + (−0.502 + 0.869i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.952 + 0.305i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.952 + 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00512268 - 0.0327290i\)
\(L(\frac12)\) \(\approx\) \(0.00512268 - 0.0327290i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0146i)T \)
good5 \( 1 + (1.45 - 2.51i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.08 - 0.628i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.32iT - 13T^{2} \)
17 \( 1 + (0.880 + 1.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.69 + 3.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.43 + 2.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.74iT - 29T^{2} \)
31 \( 1 + (-2.18 + 1.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.66 - 6.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.09T + 41T^{2} \)
43 \( 1 - 1.15T + 43T^{2} \)
47 \( 1 + (3.44 - 5.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.3 - 6.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.13 + 5.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.19 - 1.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.689 + 1.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.65iT - 71T^{2} \)
73 \( 1 + (-4.38 + 2.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.63 - 13.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.75T + 83T^{2} \)
89 \( 1 + (-3.55 + 6.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.03iT - 97T^{2} \)
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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

−11.10037311830133396065048352037, −10.62346401820799689161214493460, −9.858618249681369488257432619647, −8.650904961220627925739622163867, −7.70642797278550737118657923232, −6.81624637599728445724339130534, −6.11985085098221497831333275374, −4.68636039556424261327184303217, −3.35472777584512901439894320538, −2.68047352724672640901787581209, 0.01843287789945307748189353886, 2.01788949915396124314362925492, 3.84291344854037314404424052900, 4.39494227622066586700036367625, 5.83834910218109487143624243634, 6.64858452200421761251986828326, 7.918202839389931413073621032287, 8.638196713960344353598759553857, 9.455382708344982085739244302978, 10.32959842935929874409989796450

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