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Properties

Label 2-175-5.4-c9-0-27
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  + 14.8i·2-s − 239. i·3-s + 292.·4-s + 3.55e3·6-s − 2.40e3i·7-s + 1.19e4i·8-s − 3.77e4·9-s − 1.65e4·11-s − 7.00e4i·12-s + 2.63e4i·13-s + 3.56e4·14-s − 2.72e4·16-s + 1.44e5i·17-s − 5.60e5i·18-s + 1.59e5·19-s + ⋯
L(s)  = 1  + 0.655i·2-s − 1.70i·3-s + 0.570·4-s + 1.11·6-s − 0.377i·7-s + 1.02i·8-s − 1.91·9-s − 0.340·11-s − 0.974i·12-s + 0.255i·13-s + 0.247·14-s − 0.103·16-s + 0.418i·17-s − 1.25i·18-s + 0.281·19-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

\(L(5)\) \(\approx\) \(2.246452164\)
\(L(\frac12)\) \(\approx\) \(2.246452164\)
\(L(\frac{11}{2})\) 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 \( 1 - 14.8iT - 512T^{2} \)
3 \( 1 + 239. iT - 1.96e4T^{2} \)
11 \( 1 + 1.65e4T + 2.35e9T^{2} \)
13 \( 1 - 2.63e4iT - 1.06e10T^{2} \)
17 \( 1 - 1.44e5iT - 1.18e11T^{2} \)
19 \( 1 - 1.59e5T + 3.22e11T^{2} \)
23 \( 1 - 2.07e6iT - 1.80e12T^{2} \)
29 \( 1 - 4.94e6T + 1.45e13T^{2} \)
31 \( 1 - 4.22e6T + 2.64e13T^{2} \)
37 \( 1 - 1.29e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.87e7T + 3.27e14T^{2} \)
43 \( 1 - 3.54e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.95e7iT - 1.11e15T^{2} \)
53 \( 1 - 2.31e6iT - 3.29e15T^{2} \)
59 \( 1 - 1.68e8T + 8.66e15T^{2} \)
61 \( 1 + 6.70e7T + 1.16e16T^{2} \)
67 \( 1 - 1.56e8iT - 2.72e16T^{2} \)
71 \( 1 - 6.95e7T + 4.58e16T^{2} \)
73 \( 1 + 7.83e7iT - 5.88e16T^{2} \)
79 \( 1 - 4.26e8T + 1.19e17T^{2} \)
83 \( 1 + 5.31e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.14e8T + 3.50e17T^{2} \)
97 \( 1 - 1.46e9iT - 7.60e17T^{2} \)
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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

−11.55995211032923613848565773542, −10.19329372398641073519010588396, −8.440942829455383964489301607303, −7.82134883144711014496953838628, −6.92119247366971812386481974039, −6.33794564128637430492093249641, −5.20537749003550877479625152448, −3.09161937432092919211061403237, −1.93216098595404481962751743818, −1.03319722145540612297934964676, 0.53416562295898580710913470943, 2.41256638910707229372572248515, 3.20895386364833060935505400399, 4.34092311331886636031295134738, 5.36426541457156286212461751770, 6.62620416670885047730093130295, 8.259217413893866386339089818255, 9.316618504557064728172768295730, 10.26019652559310965512380791747, 10.68120002021342536355865471904

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