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Properties

Label 2-175-5.4-c3-0-20
Degree $2$
Conductor $175$
Sign $-0.894 - 0.447i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.62i·2-s + 8.38i·3-s − 13.3·4-s + 38.7·6-s + 7i·7-s + 24.9i·8-s − 43.3·9-s − 30.1·11-s − 112. i·12-s − 88.9i·13-s + 32.3·14-s + 8.10·16-s − 4.73i·17-s + 200. i·18-s − 124.·19-s + ⋯
L(s)  = 1  − 1.63i·2-s + 1.61i·3-s − 1.67·4-s + 2.63·6-s + 0.377i·7-s + 1.10i·8-s − 1.60·9-s − 0.825·11-s − 2.70i·12-s − 1.89i·13-s + 0.617·14-s + 0.126·16-s − 0.0675i·17-s + 2.62i·18-s − 1.50·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0500827 + 0.212153i\)
\(L(\frac12)\) \(\approx\) \(0.0500827 + 0.212153i\)
\(L(\frac{5}{2})\) 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 \( 1 - 7iT \)
good2 \( 1 + 4.62iT - 8T^{2} \)
3 \( 1 - 8.38iT - 27T^{2} \)
11 \( 1 + 30.1T + 1.33e3T^{2} \)
13 \( 1 + 88.9iT - 2.19e3T^{2} \)
17 \( 1 + 4.73iT - 4.91e3T^{2} \)
19 \( 1 + 124.T + 6.85e3T^{2} \)
23 \( 1 + 20.2iT - 1.21e4T^{2} \)
29 \( 1 + 134.T + 2.43e4T^{2} \)
31 \( 1 + 2.03T + 2.97e4T^{2} \)
37 \( 1 + 141. iT - 5.06e4T^{2} \)
41 \( 1 - 95.2T + 6.89e4T^{2} \)
43 \( 1 - 298. iT - 7.95e4T^{2} \)
47 \( 1 + 129. iT - 1.03e5T^{2} \)
53 \( 1 + 388. iT - 1.48e5T^{2} \)
59 \( 1 + 838.T + 2.05e5T^{2} \)
61 \( 1 - 389.T + 2.26e5T^{2} \)
67 \( 1 - 697. iT - 3.00e5T^{2} \)
71 \( 1 + 523.T + 3.57e5T^{2} \)
73 \( 1 + 66.4iT - 3.89e5T^{2} \)
79 \( 1 - 526.T + 4.93e5T^{2} \)
83 \( 1 + 70.0iT - 5.71e5T^{2} \)
89 \( 1 - 9.27T + 7.04e5T^{2} \)
97 \( 1 + 4.19iT - 9.12e5T^{2} \)
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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

−11.25848234198246846161341730416, −10.59893595269608799135175168528, −10.12442855057286235993870976836, −9.145763727797844815743326071983, −8.166469755608880457401211798965, −5.63643084049098432980040717033, −4.64597721459979143774824722465, −3.50975500326661890101014991270, −2.54751090868037895702696075158, −0.093442374290872021546514594534, 1.98394844495889893801492098413, 4.46690025153983776732904224606, 5.95154462912950063063656485706, 6.71864968476462883822267420060, 7.39890403130623978992935462599, 8.254258729188601394603326089598, 9.189738801868788912698725417434, 10.99420316012251071736070796260, 12.21345198670740098728044301535, 13.22805372563586425486588137603

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