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Properties

Label 2-504-9.7-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.343 - 0.939i$
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  + (−1.67 + 0.449i)3-s + (1.87 + 3.24i)5-s + (−0.5 + 0.866i)7-s + (2.59 − 1.50i)9-s + (1.82 − 3.16i)11-s + (2.77 + 4.80i)13-s + (−4.59 − 4.58i)15-s − 7.20·17-s − 3.30·19-s + (0.447 − 1.67i)21-s + (2.49 + 4.32i)23-s + (−4.52 + 7.84i)25-s + (−3.66 + 3.68i)27-s + (−0.245 + 0.425i)29-s + (1.94 + 3.37i)31-s + ⋯
L(s)  = 1  + (−0.965 + 0.259i)3-s + (0.838 + 1.45i)5-s + (−0.188 + 0.327i)7-s + (0.865 − 0.501i)9-s + (0.550 − 0.953i)11-s + (0.769 + 1.33i)13-s + (−1.18 − 1.18i)15-s − 1.74·17-s − 0.758·19-s + (0.0975 − 0.365i)21-s + (0.520 + 0.900i)23-s + (−0.905 + 1.56i)25-s + (−0.705 + 0.708i)27-s + (−0.0455 + 0.0789i)29-s + (0.349 + 0.605i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618237 + 0.884138i\)
\(L(\frac12)\) \(\approx\) \(0.618237 + 0.884138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 0.449i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.87 - 3.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.82 + 3.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.77 - 4.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.20T + 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 + (-2.49 - 4.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.245 - 0.425i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.94 - 3.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + (2.38 + 4.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.801 + 1.38i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.81 - 8.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.03T + 53T^{2} \)
59 \( 1 + (-0.754 - 1.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.04 + 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.70 + 2.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 + (1.86 - 3.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.69 + 9.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.14T + 89T^{2} \)
97 \( 1 + (-5.45 + 9.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22880769954031698110116709988, −10.57445905776945905805939366924, −9.474297138496526467315643998360, −8.858792027607682562364760774054, −7.07229044067085831131883150069, −6.28697199655147658425311089462, −6.12518382193342613915201523077, −4.53958934574829593356853980847, −3.35368252505893746334792040417, −1.88042750161605822451069998510, 0.74306173748962589421243315522, 2.02801942974619115035422819389, 4.33657229461534922571340981070, 4.89264171883953787915108814327, 6.05781140506123925365970881655, 6.64208702836634658091265541079, 8.026218620628617885173003155487, 8.930439216382318682211580023802, 9.837234009408984296926463457330, 10.62635752364429193756284925899

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