Properties

Label 2-48e2-3.2-c2-0-8
Degree $2$
Conductor $2304$
Sign $0.577 - 0.816i$
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  − 7.67i·5-s − 7.21·7-s + 6.05i·11-s − 2.29·13-s − 21.8i·17-s − 34.8·19-s + 21.5i·23-s − 33.8·25-s + 10.9i·29-s + 37.6·31-s + 55.3i·35-s + 34.8·37-s − 13.3i·41-s − 60.5·43-s + 3.34i·47-s + ⋯
L(s)  = 1  − 1.53i·5-s − 1.03·7-s + 0.550i·11-s − 0.176·13-s − 1.28i·17-s − 1.83·19-s + 0.935i·23-s − 1.35·25-s + 0.376i·29-s + 1.21·31-s + 1.58i·35-s + 0.941·37-s − 0.325i·41-s − 1.40·43-s + 0.0712i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6874242122\)
\(L(\frac12)\) \(\approx\) \(0.6874242122\)
\(L(2)\) 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 \( 1 + 7.67iT - 25T^{2} \)
7 \( 1 + 7.21T + 49T^{2} \)
11 \( 1 - 6.05iT - 121T^{2} \)
13 \( 1 + 2.29T + 169T^{2} \)
17 \( 1 + 21.8iT - 289T^{2} \)
19 \( 1 + 34.8T + 361T^{2} \)
23 \( 1 - 21.5iT - 529T^{2} \)
29 \( 1 - 10.9iT - 841T^{2} \)
31 \( 1 - 37.6T + 961T^{2} \)
37 \( 1 - 34.8T + 1.36e3T^{2} \)
41 \( 1 + 13.3iT - 1.68e3T^{2} \)
43 \( 1 + 60.5T + 1.84e3T^{2} \)
47 \( 1 - 3.34iT - 2.20e3T^{2} \)
53 \( 1 - 35.1iT - 2.80e3T^{2} \)
59 \( 1 - 37.1iT - 3.48e3T^{2} \)
61 \( 1 + 25.6T + 3.72e3T^{2} \)
67 \( 1 + 25.6T + 4.48e3T^{2} \)
71 \( 1 - 37.2iT - 5.04e3T^{2} \)
73 \( 1 - 77.6T + 5.32e3T^{2} \)
79 \( 1 + 31.3T + 6.24e3T^{2} \)
83 \( 1 + 55.3iT - 6.88e3T^{2} \)
89 \( 1 - 5.43iT - 7.92e3T^{2} \)
97 \( 1 + 52.8T + 9.40e3T^{2} \)
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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

−8.962735330035581497868680772357, −8.355498045006431755685252270505, −7.41944092874969791899845097061, −6.58598151398478708564106306674, −5.77750014835355316568960668361, −4.80203980764347546047564179482, −4.35800145788173591400202492561, −3.16809653777916044784268004572, −2.04575244351228684606999789139, −0.820434853989070251230620953222, 0.21045616311403144164334553071, 2.08292591629591784892860141045, 2.90305705142305113887467483286, 3.63025223998863736534178336211, 4.51523946146724309543378100182, 6.06570971874970174363068736979, 6.37259942318614987656873862044, 6.84566670218608083777399086157, 8.044317341581797440194882717043, 8.536088804258059679678842837743

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