Properties

Label 2-48e2-24.5-c2-0-35
Degree $2$
Conductor $2304$
Sign $0.169 + 0.985i$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
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  − 1.41·5-s − 8·7-s − 11.3·11-s − 8i·13-s + 12.7i·17-s + 32i·19-s + 33.9i·23-s − 23·25-s + 43.8·29-s + 40·31-s + 11.3·35-s + 26i·37-s − 66.4i·41-s − 16i·43-s + 11.3i·47-s + ⋯
L(s)  = 1  − 0.282·5-s − 1.14·7-s − 1.02·11-s − 0.615i·13-s + 0.748i·17-s + 1.68i·19-s + 1.47i·23-s − 0.920·25-s + 1.51·29-s + 1.29·31-s + 0.323·35-s + 0.702i·37-s − 1.62i·41-s − 0.372i·43-s + 0.240i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.169 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6527473561\)
\(L(\frac12)\) \(\approx\) \(0.6527473561\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 1.41T + 25T^{2} \)
7 \( 1 + 8T + 49T^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + 8iT - 169T^{2} \)
17 \( 1 - 12.7iT - 289T^{2} \)
19 \( 1 - 32iT - 361T^{2} \)
23 \( 1 - 33.9iT - 529T^{2} \)
29 \( 1 - 43.8T + 841T^{2} \)
31 \( 1 - 40T + 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 + 66.4iT - 1.68e3T^{2} \)
43 \( 1 + 16iT - 1.84e3T^{2} \)
47 \( 1 - 11.3iT - 2.20e3T^{2} \)
53 \( 1 + 32.5T + 2.80e3T^{2} \)
59 \( 1 + 22.6T + 3.48e3T^{2} \)
61 \( 1 + 54iT - 3.72e3T^{2} \)
67 \( 1 + 80iT - 4.48e3T^{2} \)
71 \( 1 + 79.1iT - 5.04e3T^{2} \)
73 \( 1 + 96T + 5.32e3T^{2} \)
79 \( 1 + 104T + 6.24e3T^{2} \)
83 \( 1 + 101.T + 6.88e3T^{2} \)
89 \( 1 + 77.7iT - 7.92e3T^{2} \)
97 \( 1 + 80T + 9.40e3T^{2} \)
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

−8.438721282166891288916406118309, −7.969117860047172467668598216196, −7.20975432361341106679112197268, −6.09077616887823631630613589797, −5.76904760750194826630963391705, −4.60965223868030921285037204972, −3.53437761784028678766360960763, −3.03061619013472631914891315881, −1.70451132432863807924302709996, −0.22209249059355366098538954002, 0.74745755211716692610772701923, 2.65883473144462392459839017043, 2.85974755672689855418563153033, 4.34048504619227800822847280945, 4.82685441565450504527570724668, 6.05070379627125828743973424398, 6.68668230022550861493343757827, 7.33804371451333725940538294357, 8.326122163300055193096714778621, 8.944278499296394362993241624731

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