Properties

Label 2-175-1.1-c9-0-54
Degree $2$
Conductor $175$
Sign $-1$
Analytic cond. $90.1312$
Root an. cond. $9.49374$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.17·2-s − 65.7·3-s − 427.·4-s − 602.·6-s − 2.40e3·7-s − 8.62e3·8-s − 1.53e4·9-s + 3.50e4·11-s + 2.81e4·12-s + 7.74e4·13-s − 2.20e4·14-s + 1.40e5·16-s + 2.29e5·17-s − 1.40e5·18-s + 1.64e4·19-s + 1.57e5·21-s + 3.21e5·22-s + 2.57e6·23-s + 5.66e5·24-s + 7.09e5·26-s + 2.30e6·27-s + 1.02e6·28-s − 6.62e6·29-s − 8.17e6·31-s + 5.69e6·32-s − 2.30e6·33-s + 2.10e6·34-s + ⋯
L(s)  = 1  + 0.405·2-s − 0.468·3-s − 0.835·4-s − 0.189·6-s − 0.377·7-s − 0.744·8-s − 0.780·9-s + 0.722·11-s + 0.391·12-s + 0.751·13-s − 0.153·14-s + 0.534·16-s + 0.667·17-s − 0.316·18-s + 0.0289·19-s + 0.177·21-s + 0.292·22-s + 1.91·23-s + 0.348·24-s + 0.304·26-s + 0.834·27-s + 0.315·28-s − 1.74·29-s − 1.58·31-s + 0.960·32-s − 0.338·33-s + 0.270·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(90.1312\)
Root analytic conductor: \(9.49374\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 175,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + 2.40e3T \)
good2 \( 1 - 9.17T + 512T^{2} \)
3 \( 1 + 65.7T + 1.96e4T^{2} \)
11 \( 1 - 3.50e4T + 2.35e9T^{2} \)
13 \( 1 - 7.74e4T + 1.06e10T^{2} \)
17 \( 1 - 2.29e5T + 1.18e11T^{2} \)
19 \( 1 - 1.64e4T + 3.22e11T^{2} \)
23 \( 1 - 2.57e6T + 1.80e12T^{2} \)
29 \( 1 + 6.62e6T + 1.45e13T^{2} \)
31 \( 1 + 8.17e6T + 2.64e13T^{2} \)
37 \( 1 + 9.70e6T + 1.29e14T^{2} \)
41 \( 1 - 2.98e7T + 3.27e14T^{2} \)
43 \( 1 - 1.95e7T + 5.02e14T^{2} \)
47 \( 1 + 5.93e6T + 1.11e15T^{2} \)
53 \( 1 - 2.74e7T + 3.29e15T^{2} \)
59 \( 1 - 5.24e7T + 8.66e15T^{2} \)
61 \( 1 - 2.23e7T + 1.16e16T^{2} \)
67 \( 1 + 2.74e8T + 2.72e16T^{2} \)
71 \( 1 + 3.63e8T + 4.58e16T^{2} \)
73 \( 1 + 2.09e7T + 5.88e16T^{2} \)
79 \( 1 + 2.65e8T + 1.19e17T^{2} \)
83 \( 1 - 9.43e6T + 1.86e17T^{2} \)
89 \( 1 + 6.64e8T + 3.50e17T^{2} \)
97 \( 1 - 1.20e9T + 7.60e17T^{2} \)
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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

−10.72332263277543635520212718063, −9.248424858875174770972139314890, −8.854443681950545430007021630017, −7.31599339660262361090232450626, −5.95557579400063811242387947680, −5.36274064545844210088903213003, −3.98796071192884477735336646048, −3.09407160313781715151718558873, −1.13743857261380907854333089593, 0, 1.13743857261380907854333089593, 3.09407160313781715151718558873, 3.98796071192884477735336646048, 5.36274064545844210088903213003, 5.95557579400063811242387947680, 7.31599339660262361090232450626, 8.854443681950545430007021630017, 9.248424858875174770972139314890, 10.72332263277543635520212718063

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