Properties

Label 2-1150-1.1-c1-0-25
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.618·3-s + 4-s − 0.618·6-s − 1.61·7-s − 8-s − 2.61·9-s + 3.85·11-s + 0.618·12-s − 4.09·13-s + 1.61·14-s + 16-s + 5.09·17-s + 2.61·18-s − 4.85·19-s − 1.00·21-s − 3.85·22-s − 23-s − 0.618·24-s + 4.09·26-s − 3.47·27-s − 1.61·28-s − 4.76·29-s − 2.09·31-s − 32-s + 2.38·33-s − 5.09·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.356·3-s + 0.5·4-s − 0.252·6-s − 0.611·7-s − 0.353·8-s − 0.872·9-s + 1.16·11-s + 0.178·12-s − 1.13·13-s + 0.432·14-s + 0.250·16-s + 1.23·17-s + 0.617·18-s − 1.11·19-s − 0.218·21-s − 0.821·22-s − 0.208·23-s − 0.126·24-s + 0.802·26-s − 0.668·27-s − 0.305·28-s − 0.884·29-s − 0.375·31-s − 0.176·32-s + 0.414·33-s − 0.872·34-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: \(1\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.618T + 3T^{2} \)
7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 + 4.09T + 13T^{2} \)
17 \( 1 - 5.09T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
29 \( 1 + 4.76T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 + 5.52T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 - 1.23T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 1.52T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−9.489413930475761147024393436630, −8.614865746819952084068378711126, −7.921086310117812988666562924894, −6.95892987365109291345399365340, −6.23394833996723476352050564115, −5.25058136594089464018124516345, −3.82147116423406732730973347220, −2.95640737939359410198793263623, −1.78059550635149144342141258246, 0, 1.78059550635149144342141258246, 2.95640737939359410198793263623, 3.82147116423406732730973347220, 5.25058136594089464018124516345, 6.23394833996723476352050564115, 6.95892987365109291345399365340, 7.921086310117812988666562924894, 8.614865746819952084068378711126, 9.489413930475761147024393436630

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