Properties

Label 2-175-175.109-c1-0-10
Degree $2$
Conductor $175$
Sign $0.997 - 0.0682i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.75 + 0.183i)2-s + (0.0685 − 0.322i)3-s + (1.07 + 0.228i)4-s + (2.20 + 0.357i)5-s + (0.179 − 0.552i)6-s + (−2.55 + 0.690i)7-s + (−1.51 − 0.490i)8-s + (2.64 + 1.17i)9-s + (3.79 + 1.03i)10-s + (1.99 − 0.890i)11-s + (0.147 − 0.330i)12-s + (−2.54 − 3.50i)13-s + (−4.59 + 0.739i)14-s + (0.266 − 0.687i)15-s + (−4.55 − 2.02i)16-s + (−2.97 + 2.68i)17-s + ⋯
L(s)  = 1  + (1.23 + 0.130i)2-s + (0.0395 − 0.186i)3-s + (0.536 + 0.114i)4-s + (0.987 + 0.159i)5-s + (0.0732 − 0.225i)6-s + (−0.965 + 0.261i)7-s + (−0.534 − 0.173i)8-s + (0.880 + 0.391i)9-s + (1.20 + 0.326i)10-s + (0.602 − 0.268i)11-s + (0.0425 − 0.0954i)12-s + (−0.705 − 0.971i)13-s + (−1.22 + 0.197i)14-s + (0.0688 − 0.177i)15-s + (−1.13 − 0.507i)16-s + (−0.721 + 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.997 - 0.0682i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.997 - 0.0682i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07690 + 0.0709920i\)
\(L(\frac12)\) \(\approx\) \(2.07690 + 0.0709920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.20 - 0.357i)T \)
7 \( 1 + (2.55 - 0.690i)T \)
good2 \( 1 + (-1.75 - 0.183i)T + (1.95 + 0.415i)T^{2} \)
3 \( 1 + (-0.0685 + 0.322i)T + (-2.74 - 1.22i)T^{2} \)
11 \( 1 + (-1.99 + 0.890i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (2.54 + 3.50i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.97 - 2.68i)T + (1.77 - 16.9i)T^{2} \)
19 \( 1 + (5.05 - 1.07i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (-1.36 - 0.143i)T + (22.4 + 4.78i)T^{2} \)
29 \( 1 + (-2.10 - 6.47i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.71 + 5.24i)T + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (1.79 - 4.02i)T + (-24.7 - 27.4i)T^{2} \)
41 \( 1 + (-7.67 + 5.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.649iT - 43T^{2} \)
47 \( 1 + (1.55 + 1.39i)T + (4.91 + 46.7i)T^{2} \)
53 \( 1 + (-2.44 + 11.5i)T + (-48.4 - 21.5i)T^{2} \)
59 \( 1 + (-1.02 - 9.73i)T + (-57.7 + 12.2i)T^{2} \)
61 \( 1 + (0.429 - 4.08i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-5.89 + 5.30i)T + (7.00 - 66.6i)T^{2} \)
71 \( 1 + (-1.52 - 4.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.98 - 13.4i)T + (-48.8 + 54.2i)T^{2} \)
79 \( 1 + (-9.73 + 10.8i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-1.66 - 0.539i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-0.805 + 7.66i)T + (-87.0 - 18.5i)T^{2} \)
97 \( 1 + (1.06 - 0.345i)T + (78.4 - 57.0i)T^{2} \)
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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

−12.92577263489826746205347991905, −12.39505357818652808001856787813, −10.73956235008329316379114156409, −9.814909572039377495261790823881, −8.803362151957284324691143837488, −6.96278281425107415085957935983, −6.22062427186069805570309438489, −5.21883316277218257503493403514, −3.84859429196879240210093344464, −2.42158474231403071724566716077, 2.38460112854451510750782852583, 3.99190334177290381327646866143, 4.81693096849091548827376314396, 6.34194592229052060363199220792, 6.84015995219554350052509754294, 9.216829379754783158361596112518, 9.473925858398877081537342915390, 10.83324001752502969539631025516, 12.24397376998134669273484929534, 12.77504740153446086501071253583

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