Properties

Label 2-1150-1.1-c1-0-27
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 + 3.25·3-s + 4-s + 3.25·6-s + 1.44·7-s + 8-s + 7.62·9-s − 2.03·11-s + 3.25·12-s + 0.557·13-s + 1.44·14-s + 16-s − 3.91·17-s + 7.62·18-s − 6.73·19-s + 4.70·21-s − 2.03·22-s + 23-s + 3.25·24-s + 0.557·26-s + 15.0·27-s + 1.44·28-s − 9.69·29-s − 3.07·31-s + 32-s − 6.62·33-s − 3.91·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.88·3-s + 0.5·4-s + 1.33·6-s + 0.545·7-s + 0.353·8-s + 2.54·9-s − 0.612·11-s + 0.940·12-s + 0.154·13-s + 0.385·14-s + 0.250·16-s − 0.950·17-s + 1.79·18-s − 1.54·19-s + 1.02·21-s − 0.433·22-s + 0.208·23-s + 0.665·24-s + 0.109·26-s + 2.89·27-s + 0.272·28-s − 1.80·29-s − 0.552·31-s + 0.176·32-s − 1.15·33-s − 0.672·34-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.726812972\)
\(L(\frac12)\) \(\approx\) \(4.726812972\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.25T + 3T^{2} \)
7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 + 2.03T + 11T^{2} \)
13 \( 1 - 0.557T + 13T^{2} \)
17 \( 1 + 3.91T + 17T^{2} \)
19 \( 1 + 6.73T + 19T^{2} \)
29 \( 1 + 9.69T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 - 7.03T + 41T^{2} \)
43 \( 1 + 5.06T + 43T^{2} \)
47 \( 1 - 0.659T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 - 1.73T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 9.26T + 73T^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 5.25T + 89T^{2} \)
97 \( 1 + 0.0813T + 97T^{2} \)
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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

−9.650674929738477490597755315224, −8.760230918795311277507717322905, −8.254437706222791740851655048035, −7.41172507128546138843749093288, −6.68778190371235361509665776719, −5.31053698585547357669122381418, −4.25241838966104485942741606058, −3.67111017405445143738849606348, −2.44724247530436131284952446526, −1.91124882116910577009098346391, 1.91124882116910577009098346391, 2.44724247530436131284952446526, 3.67111017405445143738849606348, 4.25241838966104485942741606058, 5.31053698585547357669122381418, 6.68778190371235361509665776719, 7.41172507128546138843749093288, 8.254437706222791740851655048035, 8.760230918795311277507717322905, 9.650674929738477490597755315224

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