Properties

Label 2-48e2-3.2-c2-0-56
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  − 7.34i·5-s + 10.3·7-s + 8.48i·11-s − 10.3·13-s + 21.2i·17-s − 20·19-s − 14.6i·23-s − 29·25-s − 36.7i·29-s + 51.9·31-s − 76.3i·35-s − 41.5·37-s − 72.1i·41-s + 40·43-s − 73.4i·47-s + ⋯
L(s)  = 1  − 1.46i·5-s + 1.48·7-s + 0.771i·11-s − 0.799·13-s + 1.24i·17-s − 1.05·19-s − 0.638i·23-s − 1.15·25-s − 1.26i·29-s + 1.67·31-s − 2.18i·35-s − 1.12·37-s − 1.75i·41-s + 0.930·43-s − 1.56i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.727177977\)
\(L(\frac12)\) \(\approx\) \(1.727177977\)
\(L(2)\) 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 \( 1 + 7.34iT - 25T^{2} \)
7 \( 1 - 10.3T + 49T^{2} \)
11 \( 1 - 8.48iT - 121T^{2} \)
13 \( 1 + 10.3T + 169T^{2} \)
17 \( 1 - 21.2iT - 289T^{2} \)
19 \( 1 + 20T + 361T^{2} \)
23 \( 1 + 14.6iT - 529T^{2} \)
29 \( 1 + 36.7iT - 841T^{2} \)
31 \( 1 - 51.9T + 961T^{2} \)
37 \( 1 + 41.5T + 1.36e3T^{2} \)
41 \( 1 + 72.1iT - 1.68e3T^{2} \)
43 \( 1 - 40T + 1.84e3T^{2} \)
47 \( 1 + 73.4iT - 2.20e3T^{2} \)
53 \( 1 + 36.7iT - 2.80e3T^{2} \)
59 \( 1 - 33.9iT - 3.48e3T^{2} \)
61 \( 1 + 3.72e3T^{2} \)
67 \( 1 + 100T + 4.48e3T^{2} \)
71 \( 1 + 73.4iT - 5.04e3T^{2} \)
73 \( 1 + 20T + 5.32e3T^{2} \)
79 \( 1 + 51.9T + 6.24e3T^{2} \)
83 \( 1 + 127. iT - 6.88e3T^{2} \)
89 \( 1 + 12.7iT - 7.92e3T^{2} \)
97 \( 1 - 40T + 9.40e3T^{2} \)
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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

−8.491478583981194866001635863850, −8.050209671028159869492767036376, −7.18806435636935367709278731178, −6.05039601933368998342131175456, −5.17322695514098306319998599112, −4.52036191493287287537996850941, −4.13727471419766797354972718425, −2.21885677719001832908850254206, −1.66382956466558677427684278933, −0.41339658558687446656873549677, 1.28980503429743713066219381100, 2.54211969231401446098746950977, 3.07243265542913663496465927939, 4.37936103391565022821156179382, 5.06096050539795166328913678520, 6.06031069412633675808882902932, 6.87858892248774014849373139489, 7.53323060303044746075082774509, 8.173035899347879244184004795842, 9.033192096656698458136883489593

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