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Properties

Label 2-468-156.71-c0-0-1
Degree $2$
Conductor $468$
Sign $0.545 + 0.838i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.22 − 1.22i)5-s + (0.707 − 0.707i)8-s + (−1.49 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (0.500 − 0.866i)16-s + (0.965 + 1.67i)17-s + (−1.67 − 0.448i)20-s + 1.99i·25-s + (−0.258 + 0.965i)26-s + (−0.448 − 0.258i)29-s + (0.258 − 0.965i)32-s + (1.36 + 1.36i)34-s + (0.5 − 0.133i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.22 − 1.22i)5-s + (0.707 − 0.707i)8-s + (−1.49 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (0.500 − 0.866i)16-s + (0.965 + 1.67i)17-s + (−1.67 − 0.448i)20-s + 1.99i·25-s + (−0.258 + 0.965i)26-s + (−0.448 − 0.258i)29-s + (0.258 − 0.965i)32-s + (1.36 + 1.36i)34-s + (0.5 − 0.133i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(0.233562\)
Root analytic conductor: \(0.483282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :0),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268786497\)
\(L(\frac12)\) \(\approx\) \(1.268786497\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - 0.517iT - T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−11.50078388935458461288470185886, −10.44959702507117397983633154294, −9.352188438424827608365128296277, −8.222731408447999371365190432701, −7.49899509588398537433577771859, −6.24740690389713154586582958628, −5.11494234378403492992173262899, −4.28373941085487304598572801077, −3.50475593461122742915986499654, −1.60796657142899555613497392353, 2.80090024452565737098268707264, 3.38716822873934351229709467686, 4.62118637118897144266460594796, 5.69195689276885355326473141435, 6.98223273965434054330010150435, 7.41894312673355941377076505696, 8.212610527362203779691531886824, 9.890073493747303673107994895866, 10.80144092127551519690279959217, 11.60108786542567031636502373411

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