Properties

Label 2-2e4-4.3-c6-0-1
Degree $2$
Conductor $16$
Sign $1$
Analytic cond. $3.68086$
Root an. cond. $1.91855$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 234·5-s + 729·9-s − 4.07e3·13-s − 990·17-s + 3.91e4·25-s − 3.18e4·29-s − 5.55e4·37-s − 8.49e4·41-s + 1.70e5·45-s + 1.17e5·49-s − 2.94e4·53-s + 2.34e5·61-s − 9.52e5·65-s + 4.27e5·73-s + 5.31e5·81-s − 2.31e5·85-s + 1.37e6·89-s − 1.47e6·97-s − 1.70e6·101-s + 4.58e5·109-s + 1.12e6·113-s − 2.96e6·117-s + ⋯
L(s)  = 1  + 1.87·5-s + 9-s − 1.85·13-s − 0.201·17-s + 2.50·25-s − 1.30·29-s − 1.09·37-s − 1.23·41-s + 1.87·45-s + 49-s − 0.197·53-s + 1.03·61-s − 3.46·65-s + 1.09·73-s + 81-s − 0.377·85-s + 1.95·89-s − 1.61·97-s − 1.65·101-s + 0.354·109-s + 0.777·113-s − 1.85·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $1$
Analytic conductor: \(3.68086\)
Root analytic conductor: \(1.91855\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: $\chi_{16} (15, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.790086510\)
\(L(\frac12)\) \(\approx\) \(1.790086510\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
5 \( 1 - 234 T + p^{6} T^{2} \)
7 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
11 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
13 \( 1 + 4070 T + p^{6} T^{2} \)
17 \( 1 + 990 T + p^{6} T^{2} \)
19 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
23 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
29 \( 1 + 31878 T + p^{6} T^{2} \)
31 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
37 \( 1 + 55510 T + p^{6} T^{2} \)
41 \( 1 + 84942 T + p^{6} T^{2} \)
43 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
47 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
53 \( 1 + 29430 T + p^{6} T^{2} \)
59 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
61 \( 1 - 234938 T + p^{6} T^{2} \)
67 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
71 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
73 \( 1 - 427570 T + p^{6} T^{2} \)
79 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
83 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
89 \( 1 - 1378962 T + p^{6} T^{2} \)
97 \( 1 + 1472510 T + p^{6} T^{2} \)
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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

−17.66711598111247337833915300720, −16.79477526492107577068652082954, −14.92961960005412272282480914790, −13.66027329831071488364659636597, −12.54111176845652332787289646763, −10.27323120633818984044199962389, −9.426182773749689790760561697792, −6.96529283450080654709598085938, −5.19821066897336074860141084941, −2.03272536499668191245491348923, 2.03272536499668191245491348923, 5.19821066897336074860141084941, 6.96529283450080654709598085938, 9.426182773749689790760561697792, 10.27323120633818984044199962389, 12.54111176845652332787289646763, 13.66027329831071488364659636597, 14.92961960005412272282480914790, 16.79477526492107577068652082954, 17.66711598111247337833915300720

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