Properties

Label 2-9900-5.4-c1-0-56
Degree $2$
Conductor $9900$
Sign $-0.447 + 0.894i$
Analytic cond. $79.0518$
Root an. cond. $8.89111$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·7-s + 11-s + 4i·13-s + 6i·17-s − 2·19-s − 4·31-s − 10i·37-s + 4i·43-s − 12i·47-s − 9·49-s + 6i·53-s + 12·59-s − 10·61-s − 4i·67-s − 8i·73-s + ⋯
L(s)  = 1  − 1.51i·7-s + 0.301·11-s + 1.10i·13-s + 1.45i·17-s − 0.458·19-s − 0.718·31-s − 1.64i·37-s + 0.609i·43-s − 1.75i·47-s − 1.28·49-s + 0.824i·53-s + 1.56·59-s − 1.28·61-s − 0.488i·67-s − 0.936i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(79.0518\)
Root analytic conductor: \(8.89111\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9900} (5149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 9900,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.220327629\)
\(L(\frac12)\) \(\approx\) \(1.220327629\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good7 \( 1 + 4iT - 7T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−7.40273294357107614680908933535, −6.77341138342921813035192399707, −6.27075962881103609538249604218, −5.41017717973094151336053205517, −4.36706647135157257728955010571, −4.03531886111023093948189265719, −3.44581037984087268675252139709, −2.09061195760863104520746843774, −1.46414766233713192781759254402, −0.29271962321137785097852004066, 1.00825236056539709944243886946, 2.20821729810326279194073856196, 2.79822483845677109786299469348, 3.47589390702759675798057411835, 4.61212301098970704146288970480, 5.24220687058520745060007070122, 5.74058301610717291777313916830, 6.47429342256373262100918634846, 7.17377824738533838674407658438, 8.029961633466090761468496123663

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