Properties

Label 2-164-41.33-c1-0-2
Degree $2$
Conductor $164$
Sign $0.650 + 0.759i$
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  + (1.21 − 1.21i)3-s + (−0.714 + 0.232i)5-s + (0.736 − 4.65i)7-s + 0.0368i·9-s + (1.45 + 2.85i)11-s + (0.0718 − 0.0113i)13-s + (−0.587 + 1.15i)15-s + (0.390 − 0.199i)17-s + (−3.87 − 0.613i)19-s + (−4.76 − 6.55i)21-s + (6.27 + 4.55i)23-s + (−3.58 + 2.60i)25-s + (3.69 + 3.69i)27-s + (−2.52 − 1.28i)29-s + (−0.161 + 0.498i)31-s + ⋯
L(s)  = 1  + (0.702 − 0.702i)3-s + (−0.319 + 0.103i)5-s + (0.278 − 1.75i)7-s + 0.0122i·9-s + (0.438 + 0.861i)11-s + (0.0199 − 0.00315i)13-s + (−0.151 + 0.297i)15-s + (0.0948 − 0.0483i)17-s + (−0.888 − 0.140i)19-s + (−1.04 − 1.43i)21-s + (1.30 + 0.950i)23-s + (−0.717 + 0.521i)25-s + (0.711 + 0.711i)27-s + (−0.469 − 0.239i)29-s + (−0.0290 + 0.0894i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(1.30954\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1/2),\ 0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23357 - 0.567414i\)
\(L(\frac12)\) \(\approx\) \(1.23357 - 0.567414i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.31 - 1.06i)T \)
good3 \( 1 + (-1.21 + 1.21i)T - 3iT^{2} \)
5 \( 1 + (0.714 - 0.232i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-0.736 + 4.65i)T + (-6.65 - 2.16i)T^{2} \)
11 \( 1 + (-1.45 - 2.85i)T + (-6.46 + 8.89i)T^{2} \)
13 \( 1 + (-0.0718 + 0.0113i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.390 + 0.199i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.87 + 0.613i)T + (18.0 + 5.87i)T^{2} \)
23 \( 1 + (-6.27 - 4.55i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.52 + 1.28i)T + (17.0 + 23.4i)T^{2} \)
31 \( 1 + (0.161 - 0.498i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.14 - 3.53i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (5.49 - 7.56i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.42 + 8.99i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.70 - 0.867i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-11.4 - 8.30i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.0743 - 0.102i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-5.60 + 11.0i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (1.14 + 2.24i)T + (-41.7 + 57.4i)T^{2} \)
73 \( 1 + 5.71iT - 73T^{2} \)
79 \( 1 + (-7.67 + 7.67i)T - 79iT^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 + (-1.05 + 6.65i)T + (-84.6 - 27.5i)T^{2} \)
97 \( 1 + (-3.29 + 6.46i)T + (-57.0 - 78.4i)T^{2} \)
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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

−13.20076911771150632535185061995, −11.69065665923992810540665493421, −10.73206034835808749732068658287, −9.696486454145324230725298195280, −8.313971277074138109914076018921, −7.38221995286677267042878337714, −6.88185097449152079051187998173, −4.74897013461386459314810045343, −3.53545675891725904714263727079, −1.60657212503846089825303092586, 2.55146695112355483790343742017, 3.84518600625954565213714686872, 5.28670869662145650400941420306, 6.48238311942942341017904534523, 8.476339157740676142423422026370, 8.650000274544156781594301258961, 9.702417562015369567131225199605, 11.06266641465494635339225935247, 11.97340469651648063565307503172, 12.85065765615390277226682322660

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