Properties

Label 2-1150-5.4-c1-0-20
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 − 1.79i·3-s − 4-s − 1.79·6-s + 2.79i·7-s + i·8-s − 0.208·9-s + 3.79·11-s + 1.79i·12-s − 1.20i·13-s + 2.79·14-s + 16-s − 3.79i·17-s + 0.208i·18-s − 1.20·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.03i·3-s − 0.5·4-s − 0.731·6-s + 1.05i·7-s + 0.353i·8-s − 0.0695·9-s + 1.14·11-s + 0.517i·12-s − 0.335i·13-s + 0.746·14-s + 0.250·16-s − 0.919i·17-s + 0.0491i·18-s − 0.277·19-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

\(L(1)\) \(\approx\) \(1.660161901\)
\(L(\frac12)\) \(\approx\) \(1.660161901\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.79iT - 3T^{2} \)
7 \( 1 - 2.79iT - 7T^{2} \)
11 \( 1 - 3.79T + 11T^{2} \)
13 \( 1 + 1.20iT - 13T^{2} \)
17 \( 1 + 3.79iT - 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 - 7.16iT - 43T^{2} \)
47 \( 1 + 13.5iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 4.41T + 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 + 7.16iT - 67T^{2} \)
71 \( 1 + 5.37T + 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 3.16T + 89T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−9.542115031487621164532454885509, −8.711167209673644749909748647696, −8.071223579601287973234763166451, −6.93770323003207233789685972182, −6.30330124635046198448482594238, −5.25400807101062211199822904272, −4.18500592664710166601502314489, −2.89234477993705233359980567544, −2.03912123681488641838359439106, −0.877650825268189882388011658252, 1.28303242384288746874006320830, 3.36338250020110385661381208095, 4.30519325339976193162150469409, 4.54224110811281379613813774736, 5.97454582131443774795944625801, 6.69246874172419418028203814061, 7.51835413098945120501715516639, 8.532002671330897523529697683309, 9.229867233457764723539480574501, 10.09922298474026099650807761649

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