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Properties

Label 2-164-41.9-c1-0-2
Degree $2$
Conductor $164$
Sign $-0.512 + 0.858i$
Analytic cond. $1.30954$
Root an. cond. $1.14435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 − 1.33i)3-s − 0.428i·5-s + (−1.90 − 1.90i)7-s + 0.571i·9-s + (−2.33 − 2.33i)11-s + (−0.571 − 0.571i)13-s + (−0.571 + 0.571i)15-s + (0.143 − 0.143i)17-s + (1.76 − 1.76i)19-s + 5.10i·21-s + 1.14·23-s + 4.81·25-s + (−3.24 + 3.24i)27-s + (1.67 + 1.67i)29-s + 8.48·31-s + ⋯
L(s)  = 1  + (−0.771 − 0.771i)3-s − 0.191i·5-s + (−0.721 − 0.721i)7-s + 0.190i·9-s + (−0.704 − 0.704i)11-s + (−0.158 − 0.158i)13-s + (−0.147 + 0.147i)15-s + (0.0349 − 0.0349i)17-s + (0.404 − 0.404i)19-s + 1.11i·21-s + 0.238·23-s + 0.963·25-s + (−0.624 + 0.624i)27-s + (0.310 + 0.310i)29-s + 1.52·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.512 + 0.858i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ -0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.345571 - 0.608876i\)
\(L(\frac12)\) \(\approx\) \(0.345571 - 0.608876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (-4.10 - 4.91i)T \)
good3 \( 1 + (1.33 + 1.33i)T + 3iT^{2} \)
5 \( 1 + 0.428iT - 5T^{2} \)
7 \( 1 + (1.90 + 1.90i)T + 7iT^{2} \)
11 \( 1 + (2.33 + 2.33i)T + 11iT^{2} \)
13 \( 1 + (0.571 + 0.571i)T + 13iT^{2} \)
17 \( 1 + (-0.143 + 0.143i)T - 17iT^{2} \)
19 \( 1 + (-1.76 + 1.76i)T - 19iT^{2} \)
23 \( 1 - 1.14T + 23T^{2} \)
29 \( 1 + (-1.67 - 1.67i)T + 29iT^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 0.917T + 37T^{2} \)
43 \( 1 + 7.16iT - 43T^{2} \)
47 \( 1 + (5.76 - 5.76i)T - 47iT^{2} \)
53 \( 1 + (9.48 + 9.48i)T + 53iT^{2} \)
59 \( 1 + 1.14T + 59T^{2} \)
61 \( 1 - 4.67iT - 61T^{2} \)
67 \( 1 + (-10.5 + 10.5i)T - 67iT^{2} \)
71 \( 1 + (-6.48 - 6.48i)T + 71iT^{2} \)
73 \( 1 + 9.48iT - 73T^{2} \)
79 \( 1 + (-0.620 - 0.620i)T + 79iT^{2} \)
83 \( 1 + 7.83T + 83T^{2} \)
89 \( 1 + (7.44 + 7.44i)T + 89iT^{2} \)
97 \( 1 + (-2.81 + 2.81i)T - 97iT^{2} \)
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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

−12.66824062508122514161279752815, −11.56577356247260462773209438407, −10.67063885972995779230461142831, −9.586906303921007527614513687145, −8.194203939246539648963757718422, −7.02863845512801265965204443660, −6.23281914106539452957218477273, −4.96793450958456983305128188646, −3.16003009389151086885538959568, −0.73491236119096723360482396841, 2.76745764755100611982080778457, 4.51820381315265579094200436953, 5.51822118283681892714289711892, 6.62370057039766040564551249589, 8.039036472291889285225797263339, 9.484057100688137481003517131895, 10.14311437670576788587689447240, 11.10044174492525385115450378194, 12.13608618780448712889864105096, 12.93590677950193926874179805906

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