Properties

Label 2-275-1.1-c5-0-32
Degree $2$
Conductor $275$
Sign $1$
Analytic cond. $44.1055$
Root an. cond. $6.64120$
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  + 10.3·2-s − 20.6·3-s + 76.0·4-s − 214.·6-s − 164.·7-s + 458.·8-s + 183.·9-s + 121·11-s − 1.57e3·12-s + 585.·13-s − 1.70e3·14-s + 2.33e3·16-s + 945.·17-s + 1.90e3·18-s + 1.14e3·19-s + 3.39e3·21-s + 1.25e3·22-s + 1.34e3·23-s − 9.46e3·24-s + 6.08e3·26-s + 1.23e3·27-s − 1.25e4·28-s + 899.·29-s − 390.·31-s + 9.55e3·32-s − 2.49e3·33-s + 9.82e3·34-s + ⋯
L(s)  = 1  + 1.83·2-s − 1.32·3-s + 2.37·4-s − 2.43·6-s − 1.26·7-s + 2.53·8-s + 0.754·9-s + 0.301·11-s − 3.14·12-s + 0.960·13-s − 2.33·14-s + 2.27·16-s + 0.793·17-s + 1.38·18-s + 0.730·19-s + 1.68·21-s + 0.554·22-s + 0.530·23-s − 3.35·24-s + 1.76·26-s + 0.325·27-s − 3.01·28-s + 0.198·29-s − 0.0730·31-s + 1.64·32-s − 0.399·33-s + 1.45·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(44.1055\)
Root analytic conductor: \(6.64120\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.057105328\)
\(L(\frac12)\) \(\approx\) \(4.057105328\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 - 121T \)
good2 \( 1 - 10.3T + 32T^{2} \)
3 \( 1 + 20.6T + 243T^{2} \)
7 \( 1 + 164.T + 1.68e4T^{2} \)
13 \( 1 - 585.T + 3.71e5T^{2} \)
17 \( 1 - 945.T + 1.41e6T^{2} \)
19 \( 1 - 1.14e3T + 2.47e6T^{2} \)
23 \( 1 - 1.34e3T + 6.43e6T^{2} \)
29 \( 1 - 899.T + 2.05e7T^{2} \)
31 \( 1 + 390.T + 2.86e7T^{2} \)
37 \( 1 - 4.47e3T + 6.93e7T^{2} \)
41 \( 1 - 1.60e4T + 1.15e8T^{2} \)
43 \( 1 - 1.99e4T + 1.47e8T^{2} \)
47 \( 1 + 1.87e3T + 2.29e8T^{2} \)
53 \( 1 + 2.35e4T + 4.18e8T^{2} \)
59 \( 1 + 3.47e4T + 7.14e8T^{2} \)
61 \( 1 - 2.57e4T + 8.44e8T^{2} \)
67 \( 1 + 5.53e4T + 1.35e9T^{2} \)
71 \( 1 - 5.68e4T + 1.80e9T^{2} \)
73 \( 1 - 4.68e4T + 2.07e9T^{2} \)
79 \( 1 + 325.T + 3.07e9T^{2} \)
83 \( 1 - 9.29e4T + 3.93e9T^{2} \)
89 \( 1 - 2.30e4T + 5.58e9T^{2} \)
97 \( 1 - 5.01e3T + 8.58e9T^{2} \)
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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

−11.29055570934900437742380116363, −10.72831678230929440862520365687, −9.466695285449491543842676573575, −7.45271588410601440274222711229, −6.32472409781639305826337106377, −6.03700286568357450502617650611, −5.05685942847644630185280020666, −3.87408513717505939821710580020, −2.92987492044587280063579488857, −0.973661936740856598300973494539, 0.973661936740856598300973494539, 2.92987492044587280063579488857, 3.87408513717505939821710580020, 5.05685942847644630185280020666, 6.03700286568357450502617650611, 6.32472409781639305826337106377, 7.45271588410601440274222711229, 9.466695285449491543842676573575, 10.72831678230929440862520365687, 11.29055570934900437742380116363

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