Properties

Label 2-39e2-1.1-c3-0-21
Degree $2$
Conductor $1521$
Sign $1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
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  − 8·4-s + 5.19·5-s − 10.3·7-s − 51.9·11-s + 64·16-s − 117·17-s − 24.2·19-s − 41.5·20-s + 18·23-s − 98·25-s + 83.1·28-s + 99·29-s − 193.·31-s − 54·35-s + 112.·37-s − 36.3·41-s + 82·43-s + 415.·44-s + 72.7·47-s − 235·49-s + 261·53-s − 270·55-s − 789.·59-s − 719·61-s − 512·64-s + 703.·67-s + 936·68-s + ⋯
L(s)  = 1  − 4-s + 0.464·5-s − 0.561·7-s − 1.42·11-s + 16-s − 1.66·17-s − 0.292·19-s − 0.464·20-s + 0.163·23-s − 0.784·25-s + 0.561·28-s + 0.633·29-s − 1.12·31-s − 0.260·35-s + 0.500·37-s − 0.138·41-s + 0.290·43-s + 1.42·44-s + 0.225·47-s − 0.685·49-s + 0.676·53-s − 0.661·55-s − 1.74·59-s − 1.50·61-s − 64-s + 1.28·67-s + 1.66·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6235436705\)
\(L(\frac12)\) \(\approx\) \(0.6235436705\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 8T^{2} \)
5 \( 1 - 5.19T + 125T^{2} \)
7 \( 1 + 10.3T + 343T^{2} \)
11 \( 1 + 51.9T + 1.33e3T^{2} \)
17 \( 1 + 117T + 4.91e3T^{2} \)
19 \( 1 + 24.2T + 6.85e3T^{2} \)
23 \( 1 - 18T + 1.21e4T^{2} \)
29 \( 1 - 99T + 2.43e4T^{2} \)
31 \( 1 + 193.T + 2.97e4T^{2} \)
37 \( 1 - 112.T + 5.06e4T^{2} \)
41 \( 1 + 36.3T + 6.89e4T^{2} \)
43 \( 1 - 82T + 7.95e4T^{2} \)
47 \( 1 - 72.7T + 1.03e5T^{2} \)
53 \( 1 - 261T + 1.48e5T^{2} \)
59 \( 1 + 789.T + 2.05e5T^{2} \)
61 \( 1 + 719T + 2.26e5T^{2} \)
67 \( 1 - 703.T + 3.00e5T^{2} \)
71 \( 1 - 467.T + 3.57e5T^{2} \)
73 \( 1 - 684.T + 3.89e5T^{2} \)
79 \( 1 + 440T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + 1.51e3T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3T + 9.12e5T^{2} \)
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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

−9.196855160486392250069417837272, −8.393941427481593426623831459224, −7.64271061413475533377782181775, −6.56612766689132293110780450780, −5.73233614353305891516211688169, −4.93660634141657034013746439986, −4.14140804317027894772438440768, −3.01002074647692909532497695576, −2.00637084948877784728035379564, −0.36853556180479021846945288883, 0.36853556180479021846945288883, 2.00637084948877784728035379564, 3.01002074647692909532497695576, 4.14140804317027894772438440768, 4.93660634141657034013746439986, 5.73233614353305891516211688169, 6.56612766689132293110780450780, 7.64271061413475533377782181775, 8.393941427481593426623831459224, 9.196855160486392250069417837272

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