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Properties

Label 2-504-63.41-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.994 + 0.105i$
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  + (−0.233 − 1.71i)3-s + (−0.0868 + 0.150i)5-s + (−2.60 + 0.486i)7-s + (−2.89 + 0.799i)9-s + (3.25 − 1.87i)11-s + (−3.54 − 2.04i)13-s + (0.278 + 0.114i)15-s − 6.00·17-s − 6.26i·19-s + (1.44 + 4.34i)21-s + (−6.37 − 3.68i)23-s + (2.48 + 4.30i)25-s + (2.04 + 4.77i)27-s + (−8.58 + 4.95i)29-s + (3.76 + 2.17i)31-s + ⋯
L(s)  = 1  + (−0.134 − 0.990i)3-s + (−0.0388 + 0.0672i)5-s + (−0.982 + 0.184i)7-s + (−0.963 + 0.266i)9-s + (0.980 − 0.566i)11-s + (−0.982 − 0.567i)13-s + (0.0718 + 0.0294i)15-s − 1.45·17-s − 1.43i·19-s + (0.314 + 0.949i)21-s + (−1.32 − 0.767i)23-s + (0.496 + 0.860i)25-s + (0.393 + 0.919i)27-s + (−1.59 + 0.920i)29-s + (0.676 + 0.390i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.994 + 0.105i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0285168 - 0.537811i\)
\(L(\frac12)\) \(\approx\) \(0.0285168 - 0.537811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.233 + 1.71i)T \)
7 \( 1 + (2.60 - 0.486i)T \)
good5 \( 1 + (0.0868 - 0.150i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.25 + 1.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.54 + 2.04i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.00T + 17T^{2} \)
19 \( 1 + 6.26iT - 19T^{2} \)
23 \( 1 + (6.37 + 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.58 - 4.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.76 - 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 + (0.489 - 0.848i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0468 - 0.0811i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.86 + 3.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.61iT - 53T^{2} \)
59 \( 1 + (-0.620 + 1.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.45 - 4.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.21 + 7.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + 15.4iT - 73T^{2} \)
79 \( 1 + (1.21 + 2.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.16 - 5.47i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-5.02 + 2.90i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68555996436532027247597634853, −9.338642609965590045997146805440, −8.828168226509457610399914378381, −7.58850210860621649605485821270, −6.70387087757283452442620595600, −6.17795636619968966825043889955, −4.88004771400862255254137427618, −3.31973964387625514361828000945, −2.22849330691342522244675733254, −0.30288662029801301917592388230, 2.34060772880096061178586168453, 3.93924198868731460740493474192, 4.33247266228511180079703382577, 5.85985093290122254076031310619, 6.56755446889878189888152999906, 7.77981295548243611472807150482, 9.040078429517097475821393174330, 9.698886078633618710249017215984, 10.13958124699828298308043488185, 11.41022381802690414563803266259

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