Properties

Label 2-1075-1.1-c5-0-114
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  − 8.90·2-s − 30.7·3-s + 47.3·4-s + 274.·6-s + 85.8·7-s − 136.·8-s + 704.·9-s − 38.5·11-s − 1.45e3·12-s + 176.·13-s − 764.·14-s − 295.·16-s − 548.·17-s − 6.27e3·18-s − 47.1·19-s − 2.64e3·21-s + 343.·22-s + 3.00e3·23-s + 4.21e3·24-s − 1.57e3·26-s − 1.41e4·27-s + 4.06e3·28-s + 1.08e3·29-s + 3.69e3·31-s + 7.01e3·32-s + 1.18e3·33-s + 4.88e3·34-s + ⋯
L(s)  = 1  − 1.57·2-s − 1.97·3-s + 1.48·4-s + 3.10·6-s + 0.661·7-s − 0.756·8-s + 2.89·9-s − 0.0959·11-s − 2.92·12-s + 0.289·13-s − 1.04·14-s − 0.288·16-s − 0.459·17-s − 4.56·18-s − 0.0299·19-s − 1.30·21-s + 0.151·22-s + 1.18·23-s + 1.49·24-s − 0.455·26-s − 3.74·27-s + 0.979·28-s + 0.239·29-s + 0.689·31-s + 1.21·32-s + 0.189·33-s + 0.724·34-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: \(0\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6436273620\)
\(L(\frac12)\) \(\approx\) \(0.6436273620\)
\(L(\frac{7}{2})\) 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 \( 1 + 8.90T + 32T^{2} \)
3 \( 1 + 30.7T + 243T^{2} \)
7 \( 1 - 85.8T + 1.68e4T^{2} \)
11 \( 1 + 38.5T + 1.61e5T^{2} \)
13 \( 1 - 176.T + 3.71e5T^{2} \)
17 \( 1 + 548.T + 1.41e6T^{2} \)
19 \( 1 + 47.1T + 2.47e6T^{2} \)
23 \( 1 - 3.00e3T + 6.43e6T^{2} \)
29 \( 1 - 1.08e3T + 2.05e7T^{2} \)
31 \( 1 - 3.69e3T + 2.86e7T^{2} \)
37 \( 1 - 1.18e4T + 6.93e7T^{2} \)
41 \( 1 - 8.99e3T + 1.15e8T^{2} \)
47 \( 1 - 9.71e3T + 2.29e8T^{2} \)
53 \( 1 - 3.18e4T + 4.18e8T^{2} \)
59 \( 1 - 8.13e3T + 7.14e8T^{2} \)
61 \( 1 - 5.02e4T + 8.44e8T^{2} \)
67 \( 1 - 2.21e3T + 1.35e9T^{2} \)
71 \( 1 + 2.31e4T + 1.80e9T^{2} \)
73 \( 1 - 8.60e4T + 2.07e9T^{2} \)
79 \( 1 + 1.96e4T + 3.07e9T^{2} \)
83 \( 1 + 3.49e4T + 3.93e9T^{2} \)
89 \( 1 - 9.97e4T + 5.58e9T^{2} \)
97 \( 1 + 6.35e4T + 8.58e9T^{2} \)
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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

−9.344258031132375499794944991543, −8.338124183109893359101443784994, −7.42434642585911917438087728122, −6.80052761210416021042465007005, −5.99952978457030363017982310870, −5.02570422096808302432306540925, −4.24879457295960388981541995911, −2.19881587267941239771941344540, −1.03511510464566213996873534658, −0.65791705438767760219235838213, 0.65791705438767760219235838213, 1.03511510464566213996873534658, 2.19881587267941239771941344540, 4.24879457295960388981541995911, 5.02570422096808302432306540925, 5.99952978457030363017982310870, 6.80052761210416021042465007005, 7.42434642585911917438087728122, 8.338124183109893359101443784994, 9.344258031132375499794944991543

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