Properties

Label 2-1075-1.1-c5-0-63
Degree $2$
Conductor $1075$
Sign $1$
Analytic cond. $172.412$
Root an. cond. $13.1305$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.582·2-s − 3.05·3-s − 31.6·4-s + 1.78·6-s + 103.·7-s + 37.0·8-s − 233.·9-s − 158.·11-s + 96.8·12-s + 578.·13-s − 60.4·14-s + 991.·16-s − 253.·17-s + 136.·18-s − 3.09e3·19-s − 317.·21-s + 92.2·22-s − 4.16e3·23-s − 113.·24-s − 337.·26-s + 1.45e3·27-s − 3.28e3·28-s + 6.77e3·29-s + 6.26e3·31-s − 1.76e3·32-s + 484.·33-s + 147.·34-s + ⋯
L(s)  = 1  − 0.103·2-s − 0.196·3-s − 0.989·4-s + 0.0202·6-s + 0.799·7-s + 0.204·8-s − 0.961·9-s − 0.394·11-s + 0.194·12-s + 0.950·13-s − 0.0823·14-s + 0.968·16-s − 0.213·17-s + 0.0990·18-s − 1.96·19-s − 0.156·21-s + 0.0406·22-s − 1.64·23-s − 0.0402·24-s − 0.0978·26-s + 0.384·27-s − 0.791·28-s + 1.49·29-s + 1.17·31-s − 0.304·32-s + 0.0774·33-s + 0.0219·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(172.412\)
Root analytic conductor: \(13.1305\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1075} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8561557105\)
\(L(\frac12)\) \(\approx\) \(0.8561557105\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - 1.84e3T \)
good2 \( 1 + 0.582T + 32T^{2} \)
3 \( 1 + 3.05T + 243T^{2} \)
7 \( 1 - 103.T + 1.68e4T^{2} \)
11 \( 1 + 158.T + 1.61e5T^{2} \)
13 \( 1 - 578.T + 3.71e5T^{2} \)
17 \( 1 + 253.T + 1.41e6T^{2} \)
19 \( 1 + 3.09e3T + 2.47e6T^{2} \)
23 \( 1 + 4.16e3T + 6.43e6T^{2} \)
29 \( 1 - 6.77e3T + 2.05e7T^{2} \)
31 \( 1 - 6.26e3T + 2.86e7T^{2} \)
37 \( 1 - 3.29e3T + 6.93e7T^{2} \)
41 \( 1 + 6.15e3T + 1.15e8T^{2} \)
47 \( 1 + 8.15e3T + 2.29e8T^{2} \)
53 \( 1 - 3.04e4T + 4.18e8T^{2} \)
59 \( 1 + 4.52e4T + 7.14e8T^{2} \)
61 \( 1 + 7.25e3T + 8.44e8T^{2} \)
67 \( 1 + 1.96e4T + 1.35e9T^{2} \)
71 \( 1 + 4.81e4T + 1.80e9T^{2} \)
73 \( 1 + 4.25e4T + 2.07e9T^{2} \)
79 \( 1 + 6.51e4T + 3.07e9T^{2} \)
83 \( 1 + 7.04e4T + 3.93e9T^{2} \)
89 \( 1 - 2.96e4T + 5.58e9T^{2} \)
97 \( 1 - 8.94e4T + 8.58e9T^{2} \)
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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

−8.791680654878444401765509972757, −8.453847923536203864236948408370, −7.916671518140288177089877185570, −6.36153783491062696812641694758, −5.81400904014596846640263005881, −4.67497304525946252943363860783, −4.18884974495274213867386722715, −2.87390008314575809484114988962, −1.64852019098604589101683933277, −0.41486277107492349816275336450, 0.41486277107492349816275336450, 1.64852019098604589101683933277, 2.87390008314575809484114988962, 4.18884974495274213867386722715, 4.67497304525946252943363860783, 5.81400904014596846640263005881, 6.36153783491062696812641694758, 7.916671518140288177089877185570, 8.453847923536203864236948408370, 8.791680654878444401765509972757

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