Properties

Label 2-1150-5.4-c1-0-5
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 + 0.302i·3-s − 4-s + 0.302·6-s + 3.30i·7-s + i·8-s + 2.90·9-s − 1.69·11-s − 0.302i·12-s − 3.30i·13-s + 3.30·14-s + 16-s + 6.90i·17-s − 2.90i·18-s − 5.90·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.174i·3-s − 0.5·4-s + 0.123·6-s + 1.24i·7-s + 0.353i·8-s + 0.969·9-s − 0.511·11-s − 0.0874i·12-s − 0.916i·13-s + 0.882·14-s + 0.250·16-s + 1.67i·17-s − 0.685i·18-s − 1.35·19-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

\(L(1)\) \(\approx\) \(1.175586192\)
\(L(\frac12)\) \(\approx\) \(1.175586192\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.302iT - 3T^{2} \)
7 \( 1 - 3.30iT - 7T^{2} \)
11 \( 1 + 1.69T + 11T^{2} \)
13 \( 1 + 3.30iT - 13T^{2} \)
17 \( 1 - 6.90iT - 17T^{2} \)
19 \( 1 + 5.90T + 19T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 0.908T + 41T^{2} \)
43 \( 1 - 9.21iT - 43T^{2} \)
47 \( 1 + 2.60iT - 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 - 3.39T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 16.3T + 71T^{2} \)
73 \( 1 - 5.81iT - 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 6.30iT - 97T^{2} \)
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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

−10.23671620465695327450840818222, −9.157517242754610773777171359945, −8.462016886086835166560370080367, −7.76535527669204986831522020539, −6.39312512637828103682891808541, −5.62232521436253832620673770507, −4.66665806121396910012215287151, −3.69966725540277802712283340580, −2.58828813060652595344920831582, −1.59339858275510356301425609484, 0.51602776230605901476758683100, 2.10826978460110325798968108821, 3.81331076608123693239281415861, 4.43874567944835561252862202381, 5.35727916230254136361379841954, 6.73309595043211026452635701941, 7.05069534485786093392430640588, 7.72114658416664675656374394818, 8.798617459542333043043557003225, 9.589569004125278615822060131527

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