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Properties

Label 2-504-56.19-c1-0-3
Degree $2$
Conductor $504$
Sign $-0.948 - 0.316i$
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  + (−1.40 − 0.141i)2-s + (1.95 + 0.398i)4-s + (−1.91 + 3.32i)5-s + (1.55 + 2.13i)7-s + (−2.70 − 0.838i)8-s + (3.17 − 4.40i)10-s + (1.28 + 2.21i)11-s − 5.99·13-s + (−1.88 − 3.23i)14-s + (3.68 + 1.56i)16-s + (3.53 − 2.04i)17-s + (−2.05 − 1.18i)19-s + (−5.08 + 5.75i)20-s + (−1.48 − 3.30i)22-s + (−6.53 − 3.77i)23-s + ⋯
L(s)  = 1  + (−0.994 − 0.100i)2-s + (0.979 + 0.199i)4-s + (−0.858 + 1.48i)5-s + (0.588 + 0.808i)7-s + (−0.955 − 0.296i)8-s + (1.00 − 1.39i)10-s + (0.386 + 0.669i)11-s − 1.66·13-s + (−0.504 − 0.863i)14-s + (0.920 + 0.390i)16-s + (0.857 − 0.494i)17-s + (−0.471 − 0.271i)19-s + (−1.13 + 1.28i)20-s + (−0.317 − 0.704i)22-s + (−1.36 − 0.787i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0744192 + 0.458381i\)
\(L(\frac12)\) \(\approx\) \(0.0744192 + 0.458381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.141i)T \)
3 \( 1 \)
7 \( 1 + (-1.55 - 2.13i)T \)
good5 \( 1 + (1.91 - 3.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.28 - 2.21i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.99T + 13T^{2} \)
17 \( 1 + (-3.53 + 2.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.05 + 1.18i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.53 + 3.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.70iT - 29T^{2} \)
31 \( 1 + (-2.90 - 5.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.61 - 2.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.96iT - 41T^{2} \)
43 \( 1 + 8.06T + 43T^{2} \)
47 \( 1 + (-0.204 + 0.353i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.41 + 2.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.05 - 5.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.34 - 2.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.38 - 4.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.21iT - 71T^{2} \)
73 \( 1 + (6.65 - 3.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.89 + 5.13i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.49iT - 83T^{2} \)
89 \( 1 + (-3.35 - 1.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.20iT - 97T^{2} \)
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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

−11.35362540044511876839917184642, −10.18501319213571050692708141977, −9.857837589993434622651959180720, −8.497157355874464772930429102027, −7.69306841348085310893361349912, −7.08494498877162467203001028146, −6.16131039585126860248895432874, −4.56911841374918970283905149939, −3.00765610423405947885898609735, −2.22024951771596759875057490701, 0.37139803931469292153893497278, 1.68578687348546574552297905030, 3.70352189130767849276383281622, 4.79253753675589728018175270421, 5.88943525630896197931094291234, 7.41328454502919384290502747025, 7.88628730901595334252421228470, 8.581411670006617086536448019799, 9.580190204633199165463528762564, 10.31923769122721291932606583966

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