↑

Properties

Label 2-468-13.4-c3-0-17
Degree $2$
Conductor $468$
Sign $-0.726 - 0.687i$
Analytic cond. $27.6128$
Root an. cond. $5.25479$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 15.8i·5-s + (−1.02 + 0.592i)7-s + (−33.6 − 19.4i)11-s + (−45.4 − 11.4i)13-s + (18.4 + 31.9i)17-s + (37.1 − 21.4i)19-s + (−101. + 176. i)23-s − 127.·25-s + (29.4 − 51.0i)29-s − 77.3i·31-s + (9.41 + 16.3i)35-s + (223. + 129. i)37-s + (−146. − 84.8i)41-s + (183. + 317. i)43-s + 249. i·47-s + ⋯
L(s)  = 1  − 1.42i·5-s + (−0.0553 + 0.0319i)7-s + (−0.923 − 0.533i)11-s + (−0.969 − 0.244i)13-s + (0.263 + 0.456i)17-s + (0.449 − 0.259i)19-s + (−0.924 + 1.60i)23-s − 1.02·25-s + (0.188 − 0.327i)29-s − 0.448i·31-s + (0.0454 + 0.0787i)35-s + (0.993 + 0.573i)37-s + (−0.559 − 0.323i)41-s + (0.649 + 1.12i)43-s + 0.775i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(27.6128\)
Root analytic conductor: \(5.25479\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :3/2),\ -0.726 - 0.687i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07030542083\)
\(L(\frac12)\) \(\approx\) \(0.07030542083\)
\(L(\frac{5}{2})\) 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 \( 1 \)
13 \( 1 + (45.4 + 11.4i)T \)
good5 \( 1 + 15.8iT - 125T^{2} \)
7 \( 1 + (1.02 - 0.592i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (33.6 + 19.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-18.4 - 31.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-37.1 + 21.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (101. - 176. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-29.4 + 51.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 77.3iT - 2.97e4T^{2} \)
37 \( 1 + (-223. - 129. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (146. + 84.8i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-183. - 317. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 249. iT - 1.03e5T^{2} \)
53 \( 1 + 157.T + 1.48e5T^{2} \)
59 \( 1 + (582. - 336. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (290. + 502. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-156. - 90.4i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-449. + 259. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 982. iT - 3.89e5T^{2} \)
79 \( 1 + 1.26e3T + 4.93e5T^{2} \)
83 \( 1 - 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + (1.28e3 + 742. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (303. - 175. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−9.799445534677197748513313707615, −9.312761881980536284209167502501, −8.034995533897675181387140519694, −7.73845397119219643010300361149, −6.02000532696465499791633488153, −5.25116048662061537619052962850, −4.37689243127352482865811707834, −2.92369402609567670732409793242, −1.37946976644903414072269504557, −0.02188569498913478368651363774, 2.24500565816320684827982765834, 3.01639719232302800722448451046, 4.41761012024336528826973567116, 5.58703593255922326989111980730, 6.78528384601862803888780519191, 7.33003518245851614352577692288, 8.301088848754622724239941830411, 9.748085418164993894642124280361, 10.23355225551830918060701873897, 11.00651652138209972045370635156

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