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Properties

Label 2-1150-5.4-c1-0-9
Degree $2$
Conductor $1150$
Sign $-0.447 - 0.894i$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
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  − i·2-s + 2.68i·3-s − 4-s + 2.68·6-s + 4.59i·7-s + i·8-s − 4.22·9-s + 5.13·11-s − 2.68i·12-s − 1.22i·13-s + 4.59·14-s + 16-s + 4.68i·17-s + 4.22i·18-s + 4.59·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.55i·3-s − 0.5·4-s + 1.09·6-s + 1.73i·7-s + 0.353i·8-s − 1.40·9-s + 1.54·11-s − 0.775i·12-s − 0.338i·13-s + 1.22·14-s + 0.250·16-s + 1.13i·17-s + 0.995i·18-s + 1.05·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.505974352\)
\(L(\frac12)\) \(\approx\) \(1.505974352\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
23 \( 1 + iT \)
good3 \( 1 - 2.68iT - 3T^{2} \)
7 \( 1 - 4.59iT - 7T^{2} \)
11 \( 1 - 5.13T + 11T^{2} \)
13 \( 1 + 1.22iT - 13T^{2} \)
17 \( 1 - 4.68iT - 17T^{2} \)
19 \( 1 - 4.59T + 19T^{2} \)
29 \( 1 + 3.37T + 29T^{2} \)
31 \( 1 + 0.777T + 31T^{2} \)
37 \( 1 + 5.81iT - 37T^{2} \)
41 \( 1 + 8.50T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 6.44iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 15.6iT - 67T^{2} \)
71 \( 1 - 1.31T + 71T^{2} \)
73 \( 1 + 4.44iT - 73T^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 + 3.81iT - 83T^{2} \)
89 \( 1 + 8.93T + 89T^{2} \)
97 \( 1 - 18.0iT - 97T^{2} \)
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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

−9.888549124996550970062593026683, −9.325637678616639311497953776525, −8.930089161506577908467622121661, −8.072169349501955309585769036696, −6.33924179702683081766912297637, −5.57104508081756144770718856979, −4.80008888994289150010264806741, −3.76810244846693806199911413662, −3.13956573974367777269258061381, −1.79891088811823884226094515511, 0.71884421102320223697660578416, 1.55448698628557568965985727263, 3.37302478685458703711872654179, 4.31997709255133834035199291930, 5.53331111356693326169108466341, 6.74857464672735571389810024031, 6.96426350996208591754886861592, 7.48819829114954556410803029324, 8.433020796392872154542921413073, 9.377751216074996078190258007893

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