Properties

Label 2-48e2-1.1-c3-0-18
Degree $2$
Conductor $2304$
Sign $1$
Analytic cond. $135.940$
Root an. cond. $11.6593$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.67·5-s − 33.0·7-s + 34.6·11-s + 82.2·13-s − 97.8·17-s + 55.8·19-s − 130.·23-s − 92.7·25-s + 147.·29-s − 101.·31-s + 187.·35-s − 184.·37-s + 237.·41-s − 199.·43-s − 334.·47-s + 752.·49-s + 102.·53-s − 196.·55-s + 105.·59-s − 717.·61-s − 466.·65-s − 316.·67-s − 800.·71-s − 301.·73-s − 1.14e3·77-s + 42.8·79-s + 1.23e3·83-s + ⋯
L(s)  = 1  − 0.507·5-s − 1.78·7-s + 0.949·11-s + 1.75·13-s − 1.39·17-s + 0.674·19-s − 1.18·23-s − 0.742·25-s + 0.944·29-s − 0.586·31-s + 0.907·35-s − 0.819·37-s + 0.904·41-s − 0.708·43-s − 1.03·47-s + 2.19·49-s + 0.264·53-s − 0.481·55-s + 0.233·59-s − 1.50·61-s − 0.891·65-s − 0.577·67-s − 1.33·71-s − 0.482·73-s − 1.69·77-s + 0.0610·79-s + 1.63·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(135.940\)
Root analytic conductor: \(11.6593\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.105535973\)
\(L(\frac12)\) \(\approx\) \(1.105535973\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.67T + 125T^{2} \)
7 \( 1 + 33.0T + 343T^{2} \)
11 \( 1 - 34.6T + 1.33e3T^{2} \)
13 \( 1 - 82.2T + 2.19e3T^{2} \)
17 \( 1 + 97.8T + 4.91e3T^{2} \)
19 \( 1 - 55.8T + 6.85e3T^{2} \)
23 \( 1 + 130.T + 1.21e4T^{2} \)
29 \( 1 - 147.T + 2.43e4T^{2} \)
31 \( 1 + 101.T + 2.97e4T^{2} \)
37 \( 1 + 184.T + 5.06e4T^{2} \)
41 \( 1 - 237.T + 6.89e4T^{2} \)
43 \( 1 + 199.T + 7.95e4T^{2} \)
47 \( 1 + 334.T + 1.03e5T^{2} \)
53 \( 1 - 102.T + 1.48e5T^{2} \)
59 \( 1 - 105.T + 2.05e5T^{2} \)
61 \( 1 + 717.T + 2.26e5T^{2} \)
67 \( 1 + 316.T + 3.00e5T^{2} \)
71 \( 1 + 800.T + 3.57e5T^{2} \)
73 \( 1 + 301.T + 3.89e5T^{2} \)
79 \( 1 - 42.8T + 4.93e5T^{2} \)
83 \( 1 - 1.23e3T + 5.71e5T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + 505.T + 9.12e5T^{2} \)
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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

−8.862535066494124996590524796113, −7.927296283104273897742235124590, −6.87954968283376875293444130512, −6.35080572075739659234012624848, −5.85963954041651549585169120675, −4.34437302640048473488639577085, −3.71013249926740750311314974267, −3.12115674623909648131096333412, −1.72959040424689087341334889802, −0.46596083391684047030186343584, 0.46596083391684047030186343584, 1.72959040424689087341334889802, 3.12115674623909648131096333412, 3.71013249926740750311314974267, 4.34437302640048473488639577085, 5.85963954041651549585169120675, 6.35080572075739659234012624848, 6.87954968283376875293444130512, 7.927296283104273897742235124590, 8.862535066494124996590524796113

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