Properties

Label 2-48e2-24.5-c2-0-52
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 − 8·7-s + 11.3·11-s + 8i·13-s + 12.7i·17-s − 32i·19-s + 33.9i·23-s − 23·25-s − 43.8·29-s + 40·31-s − 11.3·35-s − 26i·37-s − 66.4i·41-s + 16i·43-s + 11.3i·47-s + ⋯
L(s)  = 1  + 0.282·5-s − 1.14·7-s + 1.02·11-s + 0.615i·13-s + 0.748i·17-s − 1.68i·19-s + 1.47i·23-s − 0.920·25-s − 1.51·29-s + 1.29·31-s − 0.323·35-s − 0.702i·37-s − 1.62i·41-s + 0.372i·43-s + 0.240i·47-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

\(L(\frac{3}{2})\) \(\approx\) \(0.03667752542\)
\(L(\frac12)\) \(\approx\) \(0.03667752542\)
\(L(2)\) 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 \( 1 + 8T + 49T^{2} \)
11 \( 1 - 11.3T + 121T^{2} \)
13 \( 1 - 8iT - 169T^{2} \)
17 \( 1 - 12.7iT - 289T^{2} \)
19 \( 1 + 32iT - 361T^{2} \)
23 \( 1 - 33.9iT - 529T^{2} \)
29 \( 1 + 43.8T + 841T^{2} \)
31 \( 1 - 40T + 961T^{2} \)
37 \( 1 + 26iT - 1.36e3T^{2} \)
41 \( 1 + 66.4iT - 1.68e3T^{2} \)
43 \( 1 - 16iT - 1.84e3T^{2} \)
47 \( 1 - 11.3iT - 2.20e3T^{2} \)
53 \( 1 - 32.5T + 2.80e3T^{2} \)
59 \( 1 - 22.6T + 3.48e3T^{2} \)
61 \( 1 - 54iT - 3.72e3T^{2} \)
67 \( 1 - 80iT - 4.48e3T^{2} \)
71 \( 1 + 79.1iT - 5.04e3T^{2} \)
73 \( 1 + 96T + 5.32e3T^{2} \)
79 \( 1 + 104T + 6.24e3T^{2} \)
83 \( 1 - 101.T + 6.88e3T^{2} \)
89 \( 1 + 77.7iT - 7.92e3T^{2} \)
97 \( 1 + 80T + 9.40e3T^{2} \)
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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

−8.757419179723430416722830087848, −7.47208755448719434494998410274, −6.89873717083227407722932990951, −6.14647719220171940046115242553, −5.49153526887748362746881151699, −4.19511532519424435870255524045, −3.63843157890471106415011584868, −2.52443740819015770592049993572, −1.43393750777722667586831938109, −0.008980643404592978051554967899, 1.29518052050737481884793050177, 2.55977495382256667386514216578, 3.47850624896089051184017132241, 4.21997817161554947594306570506, 5.39816563331625518987754171549, 6.25373823218466162766484865047, 6.60458126636050941415000544718, 7.70341303724271353838874022206, 8.416759049353437500978055393875, 9.369773671871922505369854058771

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