↑

Properties

Label 2-504-56.19-c1-0-23
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

\(L(1)\) \(\approx\) \(2.07877 + 0.292731i\)
\(L(\frac12)\) \(\approx\) \(2.07877 + 0.292731i\)
\(L(\frac{3}{2})\) 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

−11.05042369816739840443372418105, −9.663975816227467890084881140940, −9.143984626764375530802353312186, −8.271232578896814866752020051867, −7.15340513519874055698838928503, −6.22341674182020469670265373493, −5.35359859338312053785982602043, −4.44978887076033456687603071383, −3.40907199680372204179167335753, −1.27011094732849937596963840963, 1.73028798055506482653292093669, 3.07905872942083508626177822147, 3.52759112465017588024177546205, 5.41083261125834275731104556126, 6.24974071525588921275078989901, 6.60408152760859773083242324230, 8.546649868923849611323534597374, 9.308731467383509771889564571079, 10.39798209027122112246853488343, 10.75818120512758442725491660086

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