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Properties

Label 2-175-5.4-c9-0-55
Degree $2$
Conductor $175$
Sign $0.894 - 0.447i$
Analytic cond. $90.1312$
Root an. cond. $9.49374$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 34.1i·2-s − 79.6i·3-s − 655.·4-s + 2.72e3·6-s − 2.40e3i·7-s − 4.88e3i·8-s + 1.33e4·9-s + 6.93e4·11-s + 5.21e4i·12-s + 1.05e5i·13-s + 8.20e4·14-s − 1.68e5·16-s − 5.68e5i·17-s + 4.55e5i·18-s + 3.96e5·19-s + ⋯
L(s)  = 1  + 1.50i·2-s − 0.567i·3-s − 1.27·4-s + 0.857·6-s − 0.377i·7-s − 0.421i·8-s + 0.677·9-s + 1.42·11-s + 0.726i·12-s + 1.02i·13-s + 0.570·14-s − 0.642·16-s − 1.65i·17-s + 1.02i·18-s + 0.697·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(90.1312\)
Root analytic conductor: \(9.49374\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :9/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.254255932\)
\(L(\frac12)\) \(\approx\) \(2.254255932\)
\(L(\frac{11}{2})\) 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 \( 1 + 2.40e3iT \)
good2 \( 1 - 34.1iT - 512T^{2} \)
3 \( 1 + 79.6iT - 1.96e4T^{2} \)
11 \( 1 - 6.93e4T + 2.35e9T^{2} \)
13 \( 1 - 1.05e5iT - 1.06e10T^{2} \)
17 \( 1 + 5.68e5iT - 1.18e11T^{2} \)
19 \( 1 - 3.96e5T + 3.22e11T^{2} \)
23 \( 1 + 6.20e5iT - 1.80e12T^{2} \)
29 \( 1 + 4.87e6T + 1.45e13T^{2} \)
31 \( 1 + 1.42e6T + 2.64e13T^{2} \)
37 \( 1 + 1.31e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.03e7T + 3.27e14T^{2} \)
43 \( 1 + 1.11e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.99e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.65e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.09e8T + 8.66e15T^{2} \)
61 \( 1 - 3.20e7T + 1.16e16T^{2} \)
67 \( 1 + 8.02e7iT - 2.72e16T^{2} \)
71 \( 1 - 2.07e8T + 4.58e16T^{2} \)
73 \( 1 + 2.70e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.16e8T + 1.19e17T^{2} \)
83 \( 1 + 6.82e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.47e8T + 3.50e17T^{2} \)
97 \( 1 + 1.09e9iT - 7.60e17T^{2} \)
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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

−11.30930315531984289844122070604, −9.563459830587998931612789488159, −8.964270350926035022354922639176, −7.48950489851854148530789226471, −7.07639088900920594987336060119, −6.28800810696116615971059514516, −4.95257612832066730816012836197, −3.91307490693861259471842603494, −1.89170147010155265425600322197, −0.58189767930696070154510353469, 1.06377572700568057811422880603, 1.86973545256338023267122411434, 3.45901081972234536979021788085, 3.91515692852679624164496868635, 5.29554200955262884200350155590, 6.74363773589491403564716307067, 8.348243465219400525870625098956, 9.472204149331196959047322577532, 10.02603889317069998835328758266, 10.96921134435171383595905377532

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