Properties

Label 2-1150-1.1-c3-0-80
Degree $2$
Conductor $1150$
Sign $-1$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 0.534·3-s + 4·4-s − 1.06·6-s + 28.9·7-s − 8·8-s − 26.7·9-s + 33.5·11-s + 2.13·12-s − 10.2·13-s − 57.8·14-s + 16·16-s − 56.5·17-s + 53.4·18-s − 5.26·19-s + 15.4·21-s − 67.0·22-s − 23·23-s − 4.27·24-s + 20.5·26-s − 28.7·27-s + 115.·28-s − 162.·29-s − 160.·31-s − 32·32-s + 17.9·33-s + 113.·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.102·3-s + 0.5·4-s − 0.0727·6-s + 1.56·7-s − 0.353·8-s − 0.989·9-s + 0.918·11-s + 0.0514·12-s − 0.219·13-s − 1.10·14-s + 0.250·16-s − 0.807·17-s + 0.699·18-s − 0.0635·19-s + 0.160·21-s − 0.649·22-s − 0.208·23-s − 0.0363·24-s + 0.155·26-s − 0.204·27-s + 0.781·28-s − 1.03·29-s − 0.930·31-s − 0.176·32-s + 0.0945·33-s + 0.570·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
5 \( 1 \)
23 \( 1 + 23T \)
good3 \( 1 - 0.534T + 27T^{2} \)
7 \( 1 - 28.9T + 343T^{2} \)
11 \( 1 - 33.5T + 1.33e3T^{2} \)
13 \( 1 + 10.2T + 2.19e3T^{2} \)
17 \( 1 + 56.5T + 4.91e3T^{2} \)
19 \( 1 + 5.26T + 6.85e3T^{2} \)
29 \( 1 + 162.T + 2.43e4T^{2} \)
31 \( 1 + 160.T + 2.97e4T^{2} \)
37 \( 1 - 16.9T + 5.06e4T^{2} \)
41 \( 1 - 22.7T + 6.89e4T^{2} \)
43 \( 1 + 333.T + 7.95e4T^{2} \)
47 \( 1 + 130.T + 1.03e5T^{2} \)
53 \( 1 + 673.T + 1.48e5T^{2} \)
59 \( 1 - 291.T + 2.05e5T^{2} \)
61 \( 1 - 454.T + 2.26e5T^{2} \)
67 \( 1 + 132.T + 3.00e5T^{2} \)
71 \( 1 - 121.T + 3.57e5T^{2} \)
73 \( 1 + 176.T + 3.89e5T^{2} \)
79 \( 1 - 563.T + 4.93e5T^{2} \)
83 \( 1 + 809.T + 5.71e5T^{2} \)
89 \( 1 + 702.T + 7.04e5T^{2} \)
97 \( 1 + 342.T + 9.12e5T^{2} \)
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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

−8.878192074430378630499519188429, −8.327700106319330864052862986069, −7.58707531114232559228833481455, −6.64905981156635027228985959095, −5.64895342968117731870482481246, −4.75205908310516560545933744095, −3.61807976075444855809004764366, −2.25055615071134716387833664082, −1.46854659375956602906049376366, 0, 1.46854659375956602906049376366, 2.25055615071134716387833664082, 3.61807976075444855809004764366, 4.75205908310516560545933744095, 5.64895342968117731870482481246, 6.64905981156635027228985959095, 7.58707531114232559228833481455, 8.327700106319330864052862986069, 8.878192074430378630499519188429

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