Properties

Label 2-48e2-4.3-c2-0-9
Degree $2$
Conductor $2304$
Sign $-1$
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  − 5.29·5-s + 7.48i·7-s + 5.65i·11-s − 4·13-s + 21.1·17-s + 29.9i·19-s + 22.6i·23-s + 3.00·25-s + 5.29·29-s − 22.4i·31-s − 39.5i·35-s − 28·37-s + 63.4·41-s + 29.9i·43-s − 67.8i·47-s + ⋯
L(s)  = 1  − 1.05·5-s + 1.06i·7-s + 0.514i·11-s − 0.307·13-s + 1.24·17-s + 1.57i·19-s + 0.983i·23-s + 0.120·25-s + 0.182·29-s − 0.724i·31-s − 1.13i·35-s − 0.756·37-s + 1.54·41-s + 0.696i·43-s − 1.44i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7675133554\)
\(L(\frac12)\) \(\approx\) \(0.7675133554\)
\(L(2)\) 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 \( 1 + 5.29T + 25T^{2} \)
7 \( 1 - 7.48iT - 49T^{2} \)
11 \( 1 - 5.65iT - 121T^{2} \)
13 \( 1 + 4T + 169T^{2} \)
17 \( 1 - 21.1T + 289T^{2} \)
19 \( 1 - 29.9iT - 361T^{2} \)
23 \( 1 - 22.6iT - 529T^{2} \)
29 \( 1 - 5.29T + 841T^{2} \)
31 \( 1 + 22.4iT - 961T^{2} \)
37 \( 1 + 28T + 1.36e3T^{2} \)
41 \( 1 - 63.4T + 1.68e3T^{2} \)
43 \( 1 - 29.9iT - 1.84e3T^{2} \)
47 \( 1 + 67.8iT - 2.20e3T^{2} \)
53 \( 1 - 47.6T + 2.80e3T^{2} \)
59 \( 1 - 101. iT - 3.48e3T^{2} \)
61 \( 1 + 76T + 3.72e3T^{2} \)
67 \( 1 + 59.8iT - 4.48e3T^{2} \)
71 \( 1 - 90.5iT - 5.04e3T^{2} \)
73 \( 1 + 26T + 5.32e3T^{2} \)
79 \( 1 - 127. iT - 6.24e3T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 + 42.3T + 7.92e3T^{2} \)
97 \( 1 - 18T + 9.40e3T^{2} \)
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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

−9.222425280656852042824959440135, −8.248321163854890725486490421877, −7.77647170958900311838092198189, −7.11823979730932716233649446437, −5.82908937713698727993330295431, −5.47461825129319059647224827109, −4.25045981991544227666341731161, −3.57795477640566694159895759024, −2.55798766956239175810603131567, −1.38689434686113213569854057134, 0.22558777061047336091800918224, 1.03657755234955879368478042388, 2.72274782627156850661667108940, 3.56486009815332256011269847509, 4.33943079033002599979898731955, 5.06822274678705904002383288159, 6.21677256639408718610722215269, 7.14961684644089754871899837158, 7.55886577259424093398181607189, 8.334558300508504523010370584211

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