Properties

Label 2-912-1.1-c5-0-41
Degree $2$
Conductor $912$
Sign $1$
Analytic cond. $146.270$
Root an. cond. $12.0942$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 46.6·5-s + 58.6·7-s + 81·9-s + 125.·11-s + 351.·13-s + 419.·15-s − 885.·17-s − 361·19-s + 527.·21-s − 936.·23-s − 952.·25-s + 729·27-s + 3.11e3·29-s + 1.05e4·31-s + 1.12e3·33-s + 2.73e3·35-s + 1.06e4·37-s + 3.15e3·39-s + 1.12e4·41-s − 5.84e3·43-s + 3.77e3·45-s + 4.84e3·47-s − 1.33e4·49-s − 7.97e3·51-s − 3.66e4·53-s + 5.85e3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.833·5-s + 0.452·7-s + 0.333·9-s + 0.312·11-s + 0.576·13-s + 0.481·15-s − 0.743·17-s − 0.229·19-s + 0.261·21-s − 0.369·23-s − 0.304·25-s + 0.192·27-s + 0.688·29-s + 1.97·31-s + 0.180·33-s + 0.377·35-s + 1.27·37-s + 0.332·39-s + 1.04·41-s − 0.482·43-s + 0.277·45-s + 0.320·47-s − 0.795·49-s − 0.429·51-s − 1.79·53-s + 0.260·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(146.270\)
Root analytic conductor: \(12.0942\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.050087806\)
\(L(\frac12)\) \(\approx\) \(4.050087806\)
\(L(\frac{7}{2})\) 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 \( 1 - 9T \)
19 \( 1 + 361T \)
good5 \( 1 - 46.6T + 3.12e3T^{2} \)
7 \( 1 - 58.6T + 1.68e4T^{2} \)
11 \( 1 - 125.T + 1.61e5T^{2} \)
13 \( 1 - 351.T + 3.71e5T^{2} \)
17 \( 1 + 885.T + 1.41e6T^{2} \)
23 \( 1 + 936.T + 6.43e6T^{2} \)
29 \( 1 - 3.11e3T + 2.05e7T^{2} \)
31 \( 1 - 1.05e4T + 2.86e7T^{2} \)
37 \( 1 - 1.06e4T + 6.93e7T^{2} \)
41 \( 1 - 1.12e4T + 1.15e8T^{2} \)
43 \( 1 + 5.84e3T + 1.47e8T^{2} \)
47 \( 1 - 4.84e3T + 2.29e8T^{2} \)
53 \( 1 + 3.66e4T + 4.18e8T^{2} \)
59 \( 1 - 7.92e3T + 7.14e8T^{2} \)
61 \( 1 - 4.94e4T + 8.44e8T^{2} \)
67 \( 1 + 1.51e4T + 1.35e9T^{2} \)
71 \( 1 - 3.90e4T + 1.80e9T^{2} \)
73 \( 1 - 3.68e4T + 2.07e9T^{2} \)
79 \( 1 + 8.87e3T + 3.07e9T^{2} \)
83 \( 1 - 6.18e4T + 3.93e9T^{2} \)
89 \( 1 + 2.59e4T + 5.58e9T^{2} \)
97 \( 1 + 8.83e4T + 8.58e9T^{2} \)
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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

−9.423895697223750507609644049812, −8.452004892881718769802243846529, −7.925902211100438681406401302772, −6.61622487070853669909034336444, −6.09432820353000649001563842661, −4.85063019900995481660586362539, −4.03477325268450676351575400754, −2.76340978845520624548583303013, −1.92594665863398284872737835602, −0.904569796265308037327283552690, 0.904569796265308037327283552690, 1.92594665863398284872737835602, 2.76340978845520624548583303013, 4.03477325268450676351575400754, 4.85063019900995481660586362539, 6.09432820353000649001563842661, 6.61622487070853669909034336444, 7.925902211100438681406401302772, 8.452004892881718769802243846529, 9.423895697223750507609644049812

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