Properties

Label 2-48e2-24.5-c2-0-61
Degree $2$
Conductor $2304$
Sign $-0.985 + 0.169i$
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  + 1.41·5-s − 24i·13-s − 32.5i·17-s − 23·25-s + 1.41·29-s + 70i·37-s + 69.2i·41-s − 49·49-s − 103.·53-s + 22i·61-s − 33.9i·65-s + 96·73-s − 46i·85-s − 168. i·89-s − 144·97-s + ⋯
L(s)  = 1  + 0.282·5-s − 1.84i·13-s − 1.91i·17-s − 0.920·25-s + 0.0487·29-s + 1.89i·37-s + 1.69i·41-s − 0.999·49-s − 1.94·53-s + 0.360i·61-s − 0.522i·65-s + 1.31·73-s − 0.541i·85-s − 1.89i·89-s − 1.48·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.985 + 0.169i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ -0.985 + 0.169i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6972998128\)
\(L(\frac12)\) \(\approx\) \(0.6972998128\)
\(L(2)\) 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 \( 1 - 1.41T + 25T^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + 24iT - 169T^{2} \)
17 \( 1 + 32.5iT - 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 - 1.41T + 841T^{2} \)
31 \( 1 + 961T^{2} \)
37 \( 1 - 70iT - 1.36e3T^{2} \)
41 \( 1 - 69.2iT - 1.68e3T^{2} \)
43 \( 1 - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 103.T + 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 22iT - 3.72e3T^{2} \)
67 \( 1 - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 96T + 5.32e3T^{2} \)
79 \( 1 + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + 168. iT - 7.92e3T^{2} \)
97 \( 1 + 144T + 9.40e3T^{2} \)
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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

−8.202365239962739368080389969130, −7.898450500653818807549465135643, −6.90424808053979964882234791919, −6.08263385305934633375010398408, −5.23448687442913221682203019954, −4.64865654372599771548220061868, −3.22188906562215573384401901285, −2.76277005879972610911122811248, −1.31365068187300220348025650684, −0.16319160761093751898636623169, 1.60670615511753637354622513128, 2.18194333067412668526138470071, 3.73368506844486739178244947320, 4.17156508503479391566961883679, 5.31985803890213966103242703215, 6.18218578877772472100235833310, 6.72525936889767904456656062311, 7.69273795052874336591117886594, 8.468251183867451756198536950410, 9.254085497229185929248265822851

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