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Properties

Label 2-504-9.4-c1-0-10
Degree $2$
Conductor $504$
Sign $0.939 + 0.342i$
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  + (1.5 − 0.866i)3-s + (−1 + 1.73i)5-s + (−0.5 − 0.866i)7-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + (3 − 5.19i)13-s + 3.46i·15-s + 7·17-s + 19-s + (−1.5 − 0.866i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s − 5.19i·27-s + (2 + 3.46i)29-s + (−5 + 8.66i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.188 − 0.327i)7-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + (0.832 − 1.44i)13-s + 0.894i·15-s + 1.69·17-s + 0.229·19-s + (−0.327 − 0.188i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s − 0.999i·27-s + (0.371 + 0.643i)29-s + (−0.898 + 1.55i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85315 - 0.326761i\)
\(L(\frac12)\) \(\approx\) \(1.85315 - 0.326761i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13T + 73T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.57987172856091387269307319921, −10.16651189806828001197810344513, −8.937015850406098283432099225085, −8.073881502440707503240949903814, −7.25588369781986287338199477485, −6.64587556181450672197171308630, −5.22838268089954473354893221682, −3.48335615216255018675711907617, −3.21091617412705841993752115561, −1.35037306117712095951213360578, 1.52404278828785813517052309157, 3.27389694804162546342411492395, 4.02125640619201736529774437366, 5.14941804038973511286447557072, 6.31374737523508490450492597426, 7.68998471435857884591269073388, 8.392145895600724234114854847862, 9.208792837909785447575518841160, 9.711745103301385996854714350953, 11.05650201104948513639400187977

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