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Properties

Label 2-164-164.163-c4-0-39
Degree $2$
Conductor $164$
Sign $0.999 - 0.00125i$
Analytic cond. $16.9526$
Root an. cond. $4.11736$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.33 − 2.21i)2-s − 5.00·3-s + (6.22 + 14.7i)4-s + 38.7·5-s + (16.6 + 11.0i)6-s + 50.6·7-s + (11.8 − 62.9i)8-s − 55.9·9-s + (−129. − 85.7i)10-s + 167.·11-s + (−31.1 − 73.7i)12-s + 87.0i·13-s + (−168. − 112. i)14-s − 194.·15-s + (−178. + 183. i)16-s + 509. i·17-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s − 0.556·3-s + (0.389 + 0.921i)4-s + 1.55·5-s + (0.463 + 0.307i)6-s + 1.03·7-s + (0.184 − 0.982i)8-s − 0.690·9-s + (−1.29 − 0.857i)10-s + 1.38·11-s + (−0.216 − 0.512i)12-s + 0.515i·13-s + (−0.862 − 0.571i)14-s − 0.862·15-s + (−0.696 + 0.717i)16-s + 1.76i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00125i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.00125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.999 - 0.00125i$
Analytic conductor: \(16.9526\)
Root analytic conductor: \(4.11736\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :2),\ 0.999 - 0.00125i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.505279009\)
\(L(\frac12)\) \(\approx\) \(1.505279009\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.33 + 2.21i)T \)
41 \( 1 + (-652. + 1.54e3i)T \)
good3 \( 1 + 5.00T + 81T^{2} \)
5 \( 1 - 38.7T + 625T^{2} \)
7 \( 1 - 50.6T + 2.40e3T^{2} \)
11 \( 1 - 167.T + 1.46e4T^{2} \)
13 \( 1 - 87.0iT - 2.85e4T^{2} \)
17 \( 1 - 509. iT - 8.35e4T^{2} \)
19 \( 1 + 346.T + 1.30e5T^{2} \)
23 \( 1 + 887. iT - 2.79e5T^{2} \)
29 \( 1 - 788. iT - 7.07e5T^{2} \)
31 \( 1 - 511. iT - 9.23e5T^{2} \)
37 \( 1 - 880.T + 1.87e6T^{2} \)
43 \( 1 + 966. iT - 3.41e6T^{2} \)
47 \( 1 - 1.47e3T + 4.87e6T^{2} \)
53 \( 1 - 5.11e3iT - 7.89e6T^{2} \)
59 \( 1 - 3.55e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.42e3T + 1.38e7T^{2} \)
67 \( 1 - 3.52e3T + 2.01e7T^{2} \)
71 \( 1 + 1.51e3T + 2.54e7T^{2} \)
73 \( 1 + 7.33e3T + 2.83e7T^{2} \)
79 \( 1 - 7.74e3T + 3.89e7T^{2} \)
83 \( 1 + 3.44e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.34e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.16e4iT - 8.85e7T^{2} \)
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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

−11.99215728445795311384729235057, −10.84872624962754770302445589261, −10.42298369242539964763907906379, −8.957157725283508569839668587279, −8.572957185418219098401777924647, −6.69195476510543786705304711146, −5.94320219811809732499452348933, −4.28214832725492574462898232147, −2.22859318743687798092027930555, −1.28509876018072359009408396647, 0.954943589237459346939065709771, 2.23094889027586708235490901499, 5.02292316835867538252777522706, 5.81971331268643092063724204025, 6.69957796225168403915471121819, 8.089881702569235152488759081124, 9.273001543752847431558739410951, 9.815730280398054514708071986214, 11.22865686561292395534190701923, 11.58876047236405415289659890014

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