↑

Properties

Label 2-504-56.19-c1-0-29
Degree $2$
Conductor $504$
Sign $0.830 + 0.557i$
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  + (1.40 − 0.139i)2-s + (1.96 − 0.393i)4-s + (−0.128 + 0.222i)5-s + (0.623 − 2.57i)7-s + (2.70 − 0.828i)8-s + (−0.149 + 0.331i)10-s + (−1.79 − 3.10i)11-s + 4.57·13-s + (0.518 − 3.70i)14-s + (3.68 − 1.54i)16-s + (−6.92 + 3.99i)17-s + (0.201 + 0.116i)19-s + (−0.164 + 0.487i)20-s + (−2.95 − 4.12i)22-s + (5.76 + 3.32i)23-s + ⋯
L(s)  = 1  + (0.995 − 0.0989i)2-s + (0.980 − 0.196i)4-s + (−0.0575 + 0.0996i)5-s + (0.235 − 0.971i)7-s + (0.956 − 0.293i)8-s + (−0.0474 + 0.104i)10-s + (−0.540 − 0.936i)11-s + 1.27·13-s + (0.138 − 0.990i)14-s + (0.922 − 0.386i)16-s + (−1.68 + 0.970i)17-s + (0.0463 + 0.0267i)19-s + (−0.0367 + 0.109i)20-s + (−0.631 − 0.878i)22-s + (1.20 + 0.693i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.830 + 0.557i$
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: \((2,\ 504,\ (\ :1/2),\ 0.830 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.56923 - 0.782107i\)
\(L(\frac12)\) \(\approx\) \(2.56923 - 0.782107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.139i)T \)
3 \( 1 \)
7 \( 1 + (-0.623 + 2.57i)T \)
good5 \( 1 + (0.128 - 0.222i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.79 + 3.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.57T + 13T^{2} \)
17 \( 1 + (6.92 - 3.99i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.201 - 0.116i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.76 - 3.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.80iT - 29T^{2} \)
31 \( 1 + (-1.03 - 1.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.46 + 3.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 + 5.42T + 43T^{2} \)
47 \( 1 + (1.42 - 2.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.93 - 1.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 - 1.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.44 - 7.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.867 + 1.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.97iT - 71T^{2} \)
73 \( 1 + (6.57 - 3.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.51 - 4.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.79iT - 83T^{2} \)
89 \( 1 + (-2.25 - 1.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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

−10.92382948319410320080421278260, −10.53831176972592635074600672873, −8.956023353847936317542508174744, −7.992545773346969295064076238284, −6.92894031386174088510948549925, −6.17343476190954116114845006146, −5.06260304584336680259568940516, −3.98667688791880909635287925575, −3.15288675027210460091643797716, −1.44315881813964390068205640986, 2.01579894953071319726067350055, 3.05258453585299802937209183760, 4.56746182039300968530969432200, 5.11757903704781728901453027904, 6.36955166516934129143043325804, 7.03415069014453287892294372180, 8.337300088335267123210213468875, 9.025216597613151063187116703588, 10.44667039386023597384362984468, 11.19498418277358068660730803724

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