↑

Properties

Label 2-504-8.3-c2-0-14
Degree $2$
Conductor $504$
Sign $-0.935 - 0.352i$
Analytic cond. $13.7330$
Root an. cond. $3.70580$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.59 + 1.20i)2-s + (1.11 + 3.84i)4-s + 0.769i·5-s + 2.64i·7-s + (−2.82 + 7.48i)8-s + (−0.923 + 1.23i)10-s − 13.3·11-s + 9.58i·13-s + (−3.17 + 4.23i)14-s + (−13.4 + 8.58i)16-s − 16.2·17-s + 8.48·19-s + (−2.95 + 0.860i)20-s + (−21.4 − 16.0i)22-s + 12.1i·23-s + ⋯
L(s)  = 1  + (0.799 + 0.600i)2-s + (0.279 + 0.960i)4-s + 0.153i·5-s + 0.377i·7-s + (−0.352 + 0.935i)8-s + (−0.0923 + 0.123i)10-s − 1.21·11-s + 0.737i·13-s + (−0.226 + 0.302i)14-s + (−0.843 + 0.536i)16-s − 0.958·17-s + 0.446·19-s + (−0.147 + 0.0430i)20-s + (−0.974 − 0.730i)22-s + 0.530i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ -0.935 - 0.352i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.942115808\)
\(L(\frac12)\) \(\approx\) \(1.942115808\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.59 - 1.20i)T \)
3 \( 1 \)
7 \( 1 - 2.64iT \)
good5 \( 1 - 0.769iT - 25T^{2} \)
11 \( 1 + 13.3T + 121T^{2} \)
13 \( 1 - 9.58iT - 169T^{2} \)
17 \( 1 + 16.2T + 289T^{2} \)
19 \( 1 - 8.48T + 361T^{2} \)
23 \( 1 - 12.1iT - 529T^{2} \)
29 \( 1 - 24.8iT - 841T^{2} \)
31 \( 1 + 17.5iT - 961T^{2} \)
37 \( 1 - 31.2iT - 1.36e3T^{2} \)
41 \( 1 - 7.97T + 1.68e3T^{2} \)
43 \( 1 + 52.3T + 1.84e3T^{2} \)
47 \( 1 - 8.34iT - 2.20e3T^{2} \)
53 \( 1 + 74.7iT - 2.80e3T^{2} \)
59 \( 1 - 91.8T + 3.48e3T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 + 35.0T + 4.48e3T^{2} \)
71 \( 1 - 70.3iT - 5.04e3T^{2} \)
73 \( 1 - 15.8T + 5.32e3T^{2} \)
79 \( 1 + 120. iT - 6.24e3T^{2} \)
83 \( 1 - 104.T + 6.88e3T^{2} \)
89 \( 1 - 72.9T + 7.92e3T^{2} \)
97 \( 1 - 176.T + 9.40e3T^{2} \)
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

−11.35284673317089383654166709252, −10.34783100199023960911253769994, −9.073919997980697328907551798200, −8.285419515748692233459440418133, −7.27320233479824714395857568936, −6.51184897694564835877363420404, −5.39026872572379311818166849452, −4.66279253735037377395811504580, −3.32636269995603680393014000557, −2.24991317656344470038427390449, 0.55949726084823706555241115386, 2.27244093244588494781716159521, 3.31801361982764885177754676290, 4.59750171183990448181153861179, 5.32366484171764394824175276766, 6.42931057962481131839642684768, 7.47648020015058240009420558583, 8.586608089122301906318893765184, 9.753694613430712316799138976331, 10.57972073866134668722912349361

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