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Properties

Label 2-504-63.25-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.100 - 0.994i$
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  + (−1.71 + 0.222i)3-s + (0.234 − 0.405i)5-s + (0.212 + 2.63i)7-s + (2.90 − 0.764i)9-s + (0.674 + 1.16i)11-s + (−3.16 − 5.48i)13-s + (−0.311 + 0.748i)15-s + (−2.47 + 4.28i)17-s + (2.38 + 4.13i)19-s + (−0.951 − 4.48i)21-s + (−3.81 + 6.60i)23-s + (2.39 + 4.14i)25-s + (−4.81 + 1.95i)27-s + (−1.80 + 3.12i)29-s + 6.49·31-s + ⋯
L(s)  = 1  + (−0.991 + 0.128i)3-s + (0.104 − 0.181i)5-s + (0.0802 + 0.996i)7-s + (0.966 − 0.254i)9-s + (0.203 + 0.352i)11-s + (−0.877 − 1.52i)13-s + (−0.0805 + 0.193i)15-s + (−0.599 + 1.03i)17-s + (0.548 + 0.949i)19-s + (−0.207 − 0.978i)21-s + (−0.795 + 1.37i)23-s + (0.478 + 0.828i)25-s + (−0.926 + 0.377i)27-s + (−0.335 + 0.580i)29-s + 1.16·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546824 + 0.604553i\)
\(L(\frac12)\) \(\approx\) \(0.546824 + 0.604553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 0.222i)T \)
7 \( 1 + (-0.212 - 2.63i)T \)
good5 \( 1 + (-0.234 + 0.405i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.674 - 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.16 + 5.48i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.47 - 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 4.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.81 - 6.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.80 - 3.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.49T + 31T^{2} \)
37 \( 1 + (-5.24 - 9.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0251 + 0.0435i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.431 - 0.748i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (-5.84 + 10.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.87T + 59T^{2} \)
61 \( 1 - 3.74T + 61T^{2} \)
67 \( 1 + 2.64T + 67T^{2} \)
71 \( 1 + 7.04T + 71T^{2} \)
73 \( 1 + (3.30 - 5.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.17T + 79T^{2} \)
83 \( 1 + (-4.90 + 8.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.30 - 9.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.97 + 12.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.31811709568342125590949376285, −10.08393868563410405716141333116, −9.766394499535465800209170430747, −8.411312478471563156756592011775, −7.54368973837433184024431605407, −6.28370115565867986056748978952, −5.54021535662463022540802540886, −4.80940021720575285211085140411, −3.32379298074679358826794517539, −1.62280476609174464405678701937, 0.56322508351829273413024461096, 2.37074310566596318029105993367, 4.31662074999731716248227587747, 4.74062099174552493219896406651, 6.29975516946257432235001687025, 6.85814946117245291265246664942, 7.65925594406483099245551708483, 9.117508387738361134164645526812, 9.943374032504891797977607741461, 10.79006055092864890382755723928

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