Properties

Label 2-1075-1.1-c5-0-141
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.93·2-s − 29.9·3-s + 16.1·4-s + 208.·6-s − 148.·7-s + 110.·8-s + 656.·9-s + 48.2·11-s − 484.·12-s + 448.·13-s + 1.02e3·14-s − 1.27e3·16-s − 657.·17-s − 4.55e3·18-s + 2.11e3·19-s + 4.44e3·21-s − 334.·22-s + 1.05e3·23-s − 3.30e3·24-s − 3.11e3·26-s − 1.24e4·27-s − 2.39e3·28-s + 3.72e3·29-s − 960.·31-s + 5.35e3·32-s − 1.44e3·33-s + 4.56e3·34-s + ⋯
L(s)  = 1  − 1.22·2-s − 1.92·3-s + 0.504·4-s + 2.35·6-s − 1.14·7-s + 0.607·8-s + 2.70·9-s + 0.120·11-s − 0.970·12-s + 0.736·13-s + 1.40·14-s − 1.24·16-s − 0.551·17-s − 3.31·18-s + 1.34·19-s + 2.20·21-s − 0.147·22-s + 0.414·23-s − 1.16·24-s − 0.903·26-s − 3.27·27-s − 0.576·28-s + 0.823·29-s − 0.179·31-s + 0.925·32-s − 0.231·33-s + 0.676·34-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.93T + 32T^{2} \)
3 \( 1 + 29.9T + 243T^{2} \)
7 \( 1 + 148.T + 1.68e4T^{2} \)
11 \( 1 - 48.2T + 1.61e5T^{2} \)
13 \( 1 - 448.T + 3.71e5T^{2} \)
17 \( 1 + 657.T + 1.41e6T^{2} \)
19 \( 1 - 2.11e3T + 2.47e6T^{2} \)
23 \( 1 - 1.05e3T + 6.43e6T^{2} \)
29 \( 1 - 3.72e3T + 2.05e7T^{2} \)
31 \( 1 + 960.T + 2.86e7T^{2} \)
37 \( 1 + 7.29e3T + 6.93e7T^{2} \)
41 \( 1 + 9.30e3T + 1.15e8T^{2} \)
47 \( 1 + 2.82e4T + 2.29e8T^{2} \)
53 \( 1 - 5.71e3T + 4.18e8T^{2} \)
59 \( 1 + 2.69e4T + 7.14e8T^{2} \)
61 \( 1 + 1.56e4T + 8.44e8T^{2} \)
67 \( 1 - 4.38e4T + 1.35e9T^{2} \)
71 \( 1 - 8.02e4T + 1.80e9T^{2} \)
73 \( 1 + 6.21e4T + 2.07e9T^{2} \)
79 \( 1 - 5.68e4T + 3.07e9T^{2} \)
83 \( 1 + 1.49e4T + 3.93e9T^{2} \)
89 \( 1 - 2.90e3T + 5.58e9T^{2} \)
97 \( 1 + 1.07e5T + 8.58e9T^{2} \)
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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

−9.021074909130191430278849611716, −7.84854439979246729443836167700, −6.77749483458506687760496105340, −6.57782990946000460730910661181, −5.45592537930315640888943316878, −4.64915801478191049711971940487, −3.45729128310462742493245767571, −1.58889603982817699264126035242, −0.76375492935353180607735852950, 0, 0.76375492935353180607735852950, 1.58889603982817699264126035242, 3.45729128310462742493245767571, 4.64915801478191049711971940487, 5.45592537930315640888943316878, 6.57782990946000460730910661181, 6.77749483458506687760496105340, 7.84854439979246729443836167700, 9.021074909130191430278849611716

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