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Properties

Label 2-164-4.3-c2-0-24
Degree $2$
Conductor $164$
Sign $-0.763 + 0.645i$
Analytic cond. $4.46867$
Root an. cond. $2.11392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.687 − 1.87i)2-s + 4.43i·3-s + (−3.05 + 2.58i)4-s − 5.62·5-s + (8.33 − 3.05i)6-s − 6.78i·7-s + (6.94 + 3.96i)8-s − 10.6·9-s + (3.86 + 10.5i)10-s − 14.8i·11-s + (−11.4 − 13.5i)12-s + 4.74·13-s + (−12.7 + 4.66i)14-s − 24.9i·15-s + (2.66 − 15.7i)16-s − 20.0·17-s + ⋯
L(s)  = 1  + (−0.343 − 0.939i)2-s + 1.47i·3-s + (−0.763 + 0.645i)4-s − 1.12·5-s + (1.38 − 0.508i)6-s − 0.968i·7-s + (0.868 + 0.495i)8-s − 1.18·9-s + (0.386 + 1.05i)10-s − 1.35i·11-s + (−0.954 − 1.12i)12-s + 0.365·13-s + (−0.909 + 0.333i)14-s − 1.66i·15-s + (0.166 − 0.986i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(4.46867\)
Root analytic conductor: \(2.11392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 164,\ (\ :1),\ -0.763 + 0.645i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.129122 - 0.352769i\)
\(L(\frac12)\) \(\approx\) \(0.129122 - 0.352769i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.687 + 1.87i)T \)
41 \( 1 - 6.40T \)
good3 \( 1 - 4.43iT - 9T^{2} \)
5 \( 1 + 5.62T + 25T^{2} \)
7 \( 1 + 6.78iT - 49T^{2} \)
11 \( 1 + 14.8iT - 121T^{2} \)
13 \( 1 - 4.74T + 169T^{2} \)
17 \( 1 + 20.0T + 289T^{2} \)
19 \( 1 + 37.7iT - 361T^{2} \)
23 \( 1 - 21.2iT - 529T^{2} \)
29 \( 1 + 23.5T + 841T^{2} \)
31 \( 1 - 29.4iT - 961T^{2} \)
37 \( 1 + 28.0T + 1.36e3T^{2} \)
43 \( 1 + 65.5iT - 1.84e3T^{2} \)
47 \( 1 + 28.8iT - 2.20e3T^{2} \)
53 \( 1 + 70.2T + 2.80e3T^{2} \)
59 \( 1 - 26.7iT - 3.48e3T^{2} \)
61 \( 1 + 18.1T + 3.72e3T^{2} \)
67 \( 1 + 1.42iT - 4.48e3T^{2} \)
71 \( 1 - 15.8iT - 5.04e3T^{2} \)
73 \( 1 - 89.4T + 5.32e3T^{2} \)
79 \( 1 + 107. iT - 6.24e3T^{2} \)
83 \( 1 + 76.1iT - 6.88e3T^{2} \)
89 \( 1 + 154.T + 7.92e3T^{2} \)
97 \( 1 - 51.6T + 9.40e3T^{2} \)
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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

−11.60532955266026969569256790211, −11.03435708100829711266820461853, −10.57311069108174263349656162813, −9.188750806313161090082554196823, −8.604997852640328965077716552172, −7.24266841017733489977923307124, −4.97852253669946874579372620903, −3.97417559337863076656703629036, −3.32055769037846387946553407610, −0.26166240831028860465251843025, 1.86198565409034939844822626303, 4.30151415685516354812692285284, 5.92271242205031852304225499592, 6.84153175888570273106267995766, 7.82011767344198276351611577407, 8.325122752112812425211230238232, 9.614459006975805670519658126095, 11.22025097924285752255095378201, 12.42975271768914763522941244219, 12.71374516845075354815340900668

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