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Properties

Label 2-164-164.163-c4-0-1
Degree $2$
Conductor $164$
Sign $-0.713 - 0.700i$
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  + (−3.64 − 1.65i)2-s − 12.7·3-s + (10.5 + 12.0i)4-s − 20.4·5-s + (46.3 + 21.1i)6-s + 9.19·7-s + (−18.2 − 61.3i)8-s + 81.2·9-s + (74.4 + 33.8i)10-s + 123.·11-s + (−133. − 153. i)12-s − 161. i·13-s + (−33.4 − 15.2i)14-s + 260.·15-s + (−35.1 + 253. i)16-s − 238. i·17-s + ⋯
L(s)  = 1  + (−0.910 − 0.414i)2-s − 1.41·3-s + (0.656 + 0.754i)4-s − 0.817·5-s + (1.28 + 0.586i)6-s + 0.187·7-s + (−0.285 − 0.958i)8-s + 1.00·9-s + (0.744 + 0.338i)10-s + 1.02·11-s + (−0.929 − 1.06i)12-s − 0.958i·13-s + (−0.170 − 0.0777i)14-s + 1.15·15-s + (−0.137 + 0.990i)16-s − 0.823i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $-0.713 - 0.700i$
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: \((2,\ 164,\ (\ :2),\ -0.713 - 0.700i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.01923440942\)
\(L(\frac12)\) \(\approx\) \(0.01923440942\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.64 + 1.65i)T \)
41 \( 1 + (1.67e3 - 131. i)T \)
good3 \( 1 + 12.7T + 81T^{2} \)
5 \( 1 + 20.4T + 625T^{2} \)
7 \( 1 - 9.19T + 2.40e3T^{2} \)
11 \( 1 - 123.T + 1.46e4T^{2} \)
13 \( 1 + 161. iT - 2.85e4T^{2} \)
17 \( 1 + 238. iT - 8.35e4T^{2} \)
19 \( 1 - 236.T + 1.30e5T^{2} \)
23 \( 1 + 267. iT - 2.79e5T^{2} \)
29 \( 1 - 419. iT - 7.07e5T^{2} \)
31 \( 1 - 735. iT - 9.23e5T^{2} \)
37 \( 1 + 394.T + 1.87e6T^{2} \)
43 \( 1 - 3.17e3iT - 3.41e6T^{2} \)
47 \( 1 - 321.T + 4.87e6T^{2} \)
53 \( 1 + 3.56e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.64e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.73e3T + 1.38e7T^{2} \)
67 \( 1 + 3.66e3T + 2.01e7T^{2} \)
71 \( 1 + 2.83e3T + 2.54e7T^{2} \)
73 \( 1 + 561.T + 2.83e7T^{2} \)
79 \( 1 + 2.40e3T + 3.89e7T^{2} \)
83 \( 1 + 8.04e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.33e3iT - 6.27e7T^{2} \)
97 \( 1 - 7.95e3iT - 8.85e7T^{2} \)
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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

−11.90328470885988542776417748937, −11.61994059324856718378557021301, −10.71873603667461521412946023583, −9.725804065234414300321100300326, −8.426212893488058295186561113740, −7.32080010474180344301669538636, −6.35702481310062388384504043857, −4.89459943954495488974703709288, −3.35824915239575148818237696160, −1.11691488391001481658525838809, 0.01512897058128070238061380749, 1.45142281224234210743397279817, 4.13160795392988526397872150861, 5.55758509354331301070265998603, 6.52398802573410477790771198138, 7.41842120868510392590869971540, 8.666041771949782637054044206957, 9.806124448800573135871327214152, 10.91881548002766436062573210296, 11.71632758658913259656513752897

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