Properties

Label 2-48e2-8.3-c2-0-61
Degree $2$
Conductor $2304$
Sign $-0.707 + 0.707i$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
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  − 8.11i·5-s + 9.48i·7-s + 10.5·11-s + 20.9i·13-s − 4.92·17-s − 26.9·19-s − 43.7i·23-s − 40.9·25-s + 14.5i·29-s − 25.4i·31-s + 77.0·35-s + 12.9i·37-s − 50.1·41-s − 5.03·43-s − 40.8i·47-s + ⋯
L(s)  = 1  − 1.62i·5-s + 1.35i·7-s + 0.962·11-s + 1.61i·13-s − 0.289·17-s − 1.41·19-s − 1.90i·23-s − 1.63·25-s + 0.500i·29-s − 0.822i·31-s + 2.20·35-s + 0.350i·37-s − 1.22·41-s − 0.117·43-s − 0.869i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9688384016\)
\(L(\frac12)\) \(\approx\) \(0.9688384016\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 8.11iT - 25T^{2} \)
7 \( 1 - 9.48iT - 49T^{2} \)
11 \( 1 - 10.5T + 121T^{2} \)
13 \( 1 - 20.9iT - 169T^{2} \)
17 \( 1 + 4.92T + 289T^{2} \)
19 \( 1 + 26.9T + 361T^{2} \)
23 \( 1 + 43.7iT - 529T^{2} \)
29 \( 1 - 14.5iT - 841T^{2} \)
31 \( 1 + 25.4iT - 961T^{2} \)
37 \( 1 - 12.9iT - 1.36e3T^{2} \)
41 \( 1 + 50.1T + 1.68e3T^{2} \)
43 \( 1 + 5.03T + 1.84e3T^{2} \)
47 \( 1 + 40.8iT - 2.20e3T^{2} \)
53 \( 1 + 27.2iT - 2.80e3T^{2} \)
59 \( 1 + 24.0T + 3.48e3T^{2} \)
61 \( 1 + 28.9iT - 3.72e3T^{2} \)
67 \( 1 - 65.7T + 4.48e3T^{2} \)
71 \( 1 + 90.5iT - 5.04e3T^{2} \)
73 \( 1 - 127.T + 5.32e3T^{2} \)
79 \( 1 + 35.5iT - 6.24e3T^{2} \)
83 \( 1 - 101.T + 6.88e3T^{2} \)
89 \( 1 + 15.6T + 7.92e3T^{2} \)
97 \( 1 - 73.8T + 9.40e3T^{2} \)
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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

−8.641144126242277756292247090546, −8.250246834241070899075128584409, −6.61377943417023197965271904881, −6.36820905995697151200839913345, −5.19299648541856360303852584229, −4.58192411599056834969215177394, −3.91681165727402093776885562270, −2.25598138494463696018257223005, −1.68700479650629806455347407208, −0.23828766475439324093409652771, 1.19077855987142597990982206702, 2.50938415484935028289251819199, 3.57902667409893119649344269836, 3.86077169879976808699993184142, 5.21880130184474049827022984630, 6.31864125273760542583229907516, 6.74706438868782969206275220703, 7.53281032258576913223759662556, 8.044609505423430452800072044718, 9.252592103890068795260897586604

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