Properties

Label 2-504-56.19-c1-0-18
Degree $2$
Conductor $504$
Sign $0.811 - 0.584i$
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  + (−0.221 + 1.39i)2-s + (−1.90 − 0.617i)4-s + (−0.225 + 0.390i)5-s + (−0.458 − 2.60i)7-s + (1.28 − 2.52i)8-s + (−0.495 − 0.401i)10-s + (0.360 + 0.623i)11-s + 3.48·13-s + (3.74 − 0.0641i)14-s + (3.23 + 2.34i)16-s + (3.55 − 2.05i)17-s + (3.97 + 2.29i)19-s + (0.670 − 0.603i)20-s + (−0.950 + 0.365i)22-s + (−0.0459 − 0.0265i)23-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.308i)4-s + (−0.100 + 0.174i)5-s + (−0.173 − 0.984i)7-s + (0.453 − 0.891i)8-s + (−0.156 − 0.126i)10-s + (0.108 + 0.188i)11-s + 0.967·13-s + (0.999 − 0.0171i)14-s + (0.809 + 0.587i)16-s + (0.862 − 0.498i)17-s + (0.912 + 0.526i)19-s + (0.149 − 0.135i)20-s + (−0.202 + 0.0778i)22-s + (−0.00957 − 0.00552i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.811 - 0.584i$
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: \((2,\ 504,\ (\ :1/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16240 + 0.374746i\)
\(L(\frac12)\) \(\approx\) \(1.16240 + 0.374746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.221 - 1.39i)T \)
3 \( 1 \)
7 \( 1 + (0.458 + 2.60i)T \)
good5 \( 1 + (0.225 - 0.390i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.360 - 0.623i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.48T + 13T^{2} \)
17 \( 1 + (-3.55 + 2.05i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.97 - 2.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0459 + 0.0265i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + (4.58 + 7.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.51 - 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.94iT - 41T^{2} \)
43 \( 1 - 5.17T + 43T^{2} \)
47 \( 1 + (0.460 - 0.796i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.71 + 1.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.86 + 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.54 - 4.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.93 + 8.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (3.33 - 1.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.40 + 4.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.53iT - 83T^{2} \)
89 \( 1 + (-12.6 - 7.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−10.84798805442184206789234023880, −9.846869216640006408171332320587, −9.311513336520878489533420592093, −7.893205624479708429458188230285, −7.56849733494172226033470377132, −6.44963265116170072402448140767, −5.61933722981134808303739066130, −4.35079004422015777718983016083, −3.44740407784521284411448097968, −1.00811600058501416019456735895, 1.26994287988334021609301591177, 2.79544947348637504232207370769, 3.72642617438699815355912425107, 5.08491394070381722659942624811, 5.91938067107067913803570009866, 7.40731973627943744392290817868, 8.701483158390678954924064524032, 8.883323851866850879420672623151, 10.04952979143735948384619016367, 10.89550511805680808978231735626

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