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Properties

Label 2-468-468.139-c0-0-0
Degree $2$
Conductor $468$
Sign $0.815 + 0.578i$
Analytic cond. $0.233562$
Root an. cond. $0.483282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + i·3-s − 4-s + (0.5 − 0.866i)5-s + 6-s + (0.866 + 0.5i)7-s + i·8-s − 9-s + (−0.866 − 0.5i)10-s − i·12-s + 13-s + (0.5 − 0.866i)14-s + (0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s + i·18-s + ⋯
L(s)  = 1  − i·2-s + i·3-s − 4-s + (0.5 − 0.866i)5-s + 6-s + (0.866 + 0.5i)7-s + i·8-s − 9-s + (−0.866 − 0.5i)10-s − i·12-s + 13-s + (0.5 − 0.866i)14-s + (0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.815 + 0.578i$
Analytic conductor: \(0.233562\)
Root analytic conductor: \(0.483282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :0),\ 0.815 + 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8923374008\)
\(L(\frac12)\) \(\approx\) \(0.8923374008\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
13 \( 1 - T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.16688433979955126944673717641, −10.37630357525246783452432444798, −9.316209910997288121918666633286, −8.834880705389647005037880285094, −8.163609719527793505069659401211, −6.02140905357371454043173957251, −5.01236435653071880247840796160, −4.51066512011942038098555008189, −3.15051719790142554736042063170, −1.71033156098746437453562623413, 1.70887845151641075662937554124, 3.48801885920929654025838917310, 4.93627024091727876389586172556, 6.14333227930428994273728151702, 6.66226876672320456552302961374, 7.55140104358783586782057715343, 8.381789757758974918815691018768, 9.162786856038371518174802120468, 10.74798611220440157952077821979, 11.00244950233681462022343243895

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