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Properties

Label 2-1150-5.4-c1-0-22
Degree $2$
Conductor $1150$
Sign $0.894 - 0.447i$
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 + 3i·3-s − 4-s − 3·6-s − 4i·7-s − i·8-s − 6·9-s + 3·11-s − 3i·12-s − 6i·13-s + 4·14-s + 16-s − 5i·17-s − 6i·18-s + 19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 1.51i·7-s − 0.353i·8-s − 2·9-s + 0.904·11-s − 0.866i·12-s − 1.66i·13-s + 1.06·14-s + 0.250·16-s − 1.21i·17-s − 1.41i·18-s + 0.229·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.894 - 0.447i$
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: \((2,\ 1150,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.267419455\)
\(L(\frac12)\) \(\approx\) \(1.267419455\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.875686855997859995133915973413, −9.201702570472120669913231847259, −8.270307300308170402023661450228, −7.41868141999146634942015666670, −6.50045397096991209106428489389, −5.28124174604698433513304956546, −4.81131111597909614745574856745, −3.76157656209591941522380516018, −3.28054521456622117441021853685, −0.58219191301645104474559719717, 1.49255824363746871990693286365, 2.00294530278641836463803709687, 3.08057667986912756346616320090, 4.46008936653674228997610331290, 5.82556245053230439488920449592, 6.34645237528279951584791845762, 7.17302662787870385898224850780, 8.344126003375559148522213436135, 8.816594488343647122622406583828, 9.459272191888040332779141888420

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