↑

Properties

Label 2-504-56.19-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.961 + 0.276i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.826 − 1.14i)2-s + (−0.634 + 1.89i)4-s + (−1.91 + 3.32i)5-s + (−1.55 − 2.13i)7-s + (2.70 − 0.838i)8-s + (5.40 − 0.543i)10-s + (−1.28 − 2.21i)11-s + 5.99·13-s + (−1.16 + 3.55i)14-s + (−3.19 − 2.40i)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  + (−0.584 − 0.811i)2-s + (−0.317 + 0.948i)4-s + (−0.858 + 1.48i)5-s + (−0.588 − 0.808i)7-s + (0.955 − 0.296i)8-s + (1.70 − 0.171i)10-s + (−0.386 − 0.669i)11-s + 1.66·13-s + (−0.312 + 0.949i)14-s + (−0.798 − 0.602i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.961 + 0.276i$
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: \((2,\ 504,\ (\ :1/2),\ -0.961 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0361974 - 0.257049i\)
\(L(\frac12)\) \(\approx\) \(0.0361974 - 0.257049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 + 1.14i)T \)
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 \( 1 + 1.20iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73972046747562838041122399800, −10.01466108427484641517437357020, −8.614474043272166728683590434771, −8.010593369567912593629953823086, −6.94251356812853889448092114640, −6.25131721503459506682114350530, −3.86288972790066846169758775368, −3.75286852698255518507230642307, −2.38393758606856283003561288692, −0.18910289035734839812862115547, 1.59618716271673079882523797046, 3.84903274709639071745160453831, 4.91820699134073241366252577538, 5.77454536243004004557741579398, 6.81335491883286032121288108409, 8.021006191408866120842790106805, 8.617427007442468853221680279663, 9.148694105116506193318242295949, 10.15717018898714012283162020921, 11.34074794885902277336452451829

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