Properties

Label 2-48e2-24.5-c2-0-49
Degree $2$
Conductor $2304$
Sign $-0.169 + 0.985i$
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  + 4.24·5-s + 4·7-s − 16.9·11-s − 8i·13-s − 12.7i·17-s + 16i·19-s + 16.9i·23-s − 7.00·25-s + 4.24·29-s + 44·31-s + 16.9·35-s − 34i·37-s − 46.6i·41-s − 40i·43-s − 84.8i·47-s + ⋯
L(s)  = 1  + 0.848·5-s + 0.571·7-s − 1.54·11-s − 0.615i·13-s − 0.748i·17-s + 0.842i·19-s + 0.737i·23-s − 0.280·25-s + 0.146·29-s + 1.41·31-s + 0.484·35-s − 0.918i·37-s − 1.13i·41-s − 0.930i·43-s − 1.80i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.169 + 0.985i$
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: \((2,\ 2304,\ (\ :1),\ -0.169 + 0.985i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.619945190\)
\(L(\frac12)\) \(\approx\) \(1.619945190\)
\(L(2)\) 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 \( 1 - 4.24T + 25T^{2} \)
7 \( 1 - 4T + 49T^{2} \)
11 \( 1 + 16.9T + 121T^{2} \)
13 \( 1 + 8iT - 169T^{2} \)
17 \( 1 + 12.7iT - 289T^{2} \)
19 \( 1 - 16iT - 361T^{2} \)
23 \( 1 - 16.9iT - 529T^{2} \)
29 \( 1 - 4.24T + 841T^{2} \)
31 \( 1 - 44T + 961T^{2} \)
37 \( 1 + 34iT - 1.36e3T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 + 40iT - 1.84e3T^{2} \)
47 \( 1 + 84.8iT - 2.20e3T^{2} \)
53 \( 1 + 38.1T + 2.80e3T^{2} \)
59 \( 1 + 33.9T + 3.48e3T^{2} \)
61 \( 1 + 50iT - 3.72e3T^{2} \)
67 \( 1 + 8iT - 4.48e3T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 - 16T + 5.32e3T^{2} \)
79 \( 1 + 76T + 6.24e3T^{2} \)
83 \( 1 - 118.T + 6.88e3T^{2} \)
89 \( 1 + 12.7iT - 7.92e3T^{2} \)
97 \( 1 - 176T + 9.40e3T^{2} \)
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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

−8.491152781393405703592322700322, −7.87593961438020655077077989770, −7.21921754185563670911973187489, −6.06731952945753248223027695523, −5.39412280022005763078632898776, −4.91752098140652461905448100610, −3.60247648213435426086943048442, −2.57067832050492290453069938379, −1.80027665087945376187880665899, −0.37686662153160130292732900161, 1.24107438022094981731278304277, 2.29976014619415866652152706402, 3.00238115578012267474457659008, 4.60220576230541033344308361120, 4.86450393601892199271295197811, 6.06337032267329610217027752168, 6.47137477871677907972182098897, 7.74275019918113754042766961224, 8.137542000404858780304159144624, 9.058561395953441753564491470521

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