Properties

Label 2-1150-1.1-c1-0-23
Degree $2$
Conductor $1150$
Sign $1$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.79·3-s + 4-s + 2.79·6-s + 1.79·7-s + 8-s + 4.79·9-s − 0.791·11-s + 2.79·12-s − 5.79·13-s + 1.79·14-s + 16-s − 0.791·17-s + 4.79·18-s + 5.79·19-s + 5·21-s − 0.791·22-s − 23-s + 2.79·24-s − 5.79·26-s + 4.99·27-s + 1.79·28-s + 7.58·29-s − 3.37·31-s + 32-s − 2.20·33-s − 0.791·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.61·3-s + 0.5·4-s + 1.13·6-s + 0.677·7-s + 0.353·8-s + 1.59·9-s − 0.238·11-s + 0.805·12-s − 1.60·13-s + 0.478·14-s + 0.250·16-s − 0.191·17-s + 1.12·18-s + 1.32·19-s + 1.09·21-s − 0.168·22-s − 0.208·23-s + 0.569·24-s − 1.13·26-s + 0.962·27-s + 0.338·28-s + 1.40·29-s − 0.605·31-s + 0.176·32-s − 0.384·33-s − 0.135·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(4.382637652\)
\(L(\frac12)\) \(\approx\) \(4.382637652\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
23 \( 1 + T \)
good3 \( 1 - 2.79T + 3T^{2} \)
7 \( 1 - 1.79T + 7T^{2} \)
11 \( 1 + 0.791T + 11T^{2} \)
13 \( 1 + 5.79T + 13T^{2} \)
17 \( 1 + 0.791T + 17T^{2} \)
19 \( 1 - 5.79T + 19T^{2} \)
29 \( 1 - 7.58T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 4.41T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 7.95T + 97T^{2} \)
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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

−9.747972690496255641845605801117, −8.940352463843414538479869514280, −7.912786698904494383053855968820, −7.60807321389107926086352318157, −6.63688729260059636407148142703, −5.15515465410233597640554913891, −4.60694076861134357901486980124, −3.39351292271383808193163982811, −2.68347578616787567756974686316, −1.73086645546949651249292237371, 1.73086645546949651249292237371, 2.68347578616787567756974686316, 3.39351292271383808193163982811, 4.60694076861134357901486980124, 5.15515465410233597640554913891, 6.63688729260059636407148142703, 7.60807321389107926086352318157, 7.912786698904494383053855968820, 8.940352463843414538479869514280, 9.747972690496255641845605801117

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