Properties

Label 2-48e2-8.3-c2-0-11
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  + 8.92i·5-s + 10.9i·7-s + 1.07·11-s + 3.85i·13-s − 7.85·17-s − 17.0·19-s − 8i·23-s − 54.7·25-s − 3.07i·29-s + 30.6i·31-s − 97.5·35-s + 45.7i·37-s − 35.8·41-s + 74.6·43-s + 42.1i·47-s + ⋯
L(s)  = 1  + 1.78i·5-s + 1.56i·7-s + 0.0974·11-s + 0.296i·13-s − 0.462·17-s − 0.898·19-s − 0.347i·23-s − 2.18·25-s − 0.105i·29-s + 0.988i·31-s − 2.78·35-s + 1.23i·37-s − 0.874·41-s + 1.73·43-s + 0.896i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.129134524\)
\(L(\frac12)\) \(\approx\) \(1.129134524\)
\(L(2)\) 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 \( 1 - 8.92iT - 25T^{2} \)
7 \( 1 - 10.9iT - 49T^{2} \)
11 \( 1 - 1.07T + 121T^{2} \)
13 \( 1 - 3.85iT - 169T^{2} \)
17 \( 1 + 7.85T + 289T^{2} \)
19 \( 1 + 17.0T + 361T^{2} \)
23 \( 1 + 8iT - 529T^{2} \)
29 \( 1 + 3.07iT - 841T^{2} \)
31 \( 1 - 30.6iT - 961T^{2} \)
37 \( 1 - 45.7iT - 1.36e3T^{2} \)
41 \( 1 + 35.8T + 1.68e3T^{2} \)
43 \( 1 - 74.6T + 1.84e3T^{2} \)
47 \( 1 - 42.1iT - 2.20e3T^{2} \)
53 \( 1 + 12.9iT - 2.80e3T^{2} \)
59 \( 1 + 44.2T + 3.48e3T^{2} \)
61 \( 1 - 14iT - 3.72e3T^{2} \)
67 \( 1 - 80.4T + 4.48e3T^{2} \)
71 \( 1 + 123. iT - 5.04e3T^{2} \)
73 \( 1 - 85.4T + 5.32e3T^{2} \)
79 \( 1 - 55.2iT - 6.24e3T^{2} \)
83 \( 1 - 49.0T + 6.88e3T^{2} \)
89 \( 1 + 105.T + 7.92e3T^{2} \)
97 \( 1 - 21.1T + 9.40e3T^{2} \)
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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

−9.283869918485452125050567367901, −8.565274061855069824729931683672, −7.76662800539343761109849018299, −6.67276807372797871430480892383, −6.43782228495004338252579620391, −5.57896084345530217896217176549, −4.48750693164051490121283693955, −3.32640798930854011757677309729, −2.62226890445037710404204350051, −1.94799143659227697220575469748, 0.30144666633339491777063753275, 1.00673888957504010233367861787, 2.09853497482062940135184234356, 3.83367063176864826668639996525, 4.20826656435612042804264441493, 5.03411043807680727514917762741, 5.86637917389443057834166870828, 6.90262926309713914732248408572, 7.71773215414516099982785480919, 8.319068090172502989501562781641

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