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Properties

Label 2-175-35.9-c3-0-5
Degree $2$
Conductor $175$
Sign $0.0841 - 0.996i$
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  + (−4.35 − 2.51i)2-s + (−7.22 + 4.17i)3-s + (8.62 + 14.9i)4-s + 41.9·6-s + (7.81 + 16.7i)7-s − 46.5i·8-s + (21.3 − 36.9i)9-s + (0.444 + 0.769i)11-s + (−124. − 71.9i)12-s − 25.9i·13-s + (8.15 − 92.7i)14-s + (−47.8 + 82.8i)16-s + (83.4 − 48.1i)17-s + (−185. + 107. i)18-s + (−44.5 + 77.1i)19-s + ⋯
L(s)  = 1  + (−1.53 − 0.888i)2-s + (−1.39 + 0.802i)3-s + (1.07 + 1.86i)4-s + 2.85·6-s + (0.422 + 0.906i)7-s − 2.05i·8-s + (0.789 − 1.36i)9-s + (0.0121 + 0.0210i)11-s + (−2.99 − 1.73i)12-s − 0.553i·13-s + (0.155 − 1.76i)14-s + (−0.747 + 1.29i)16-s + (1.18 − 0.687i)17-s + (−2.42 + 1.40i)18-s + (−0.537 + 0.931i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.0841 - 0.996i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.0841 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.284985 + 0.261927i\)
\(L(\frac12)\) \(\approx\) \(0.284985 + 0.261927i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.81 - 16.7i)T \)
good2 \( 1 + (4.35 + 2.51i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (7.22 - 4.17i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (-0.444 - 0.769i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 25.9iT - 2.19e3T^{2} \)
17 \( 1 + (-83.4 + 48.1i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (44.5 - 77.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-100. - 58.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 222.T + 2.43e4T^{2} \)
31 \( 1 + (-6.45 - 11.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-79.0 - 45.6i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 98.4T + 6.89e4T^{2} \)
43 \( 1 - 392. iT - 7.95e4T^{2} \)
47 \( 1 + (190. + 110. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (198. - 114. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-6.76 - 11.7i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (102. - 178. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (282. - 162. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 583.T + 3.57e5T^{2} \)
73 \( 1 + (823. - 475. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-225. + 390. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 164. iT - 5.71e5T^{2} \)
89 \( 1 + (-442. + 766. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 62.1iT - 9.12e5T^{2} \)
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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

−11.87446230865902794229961742473, −11.39286530183300659128128219470, −10.36785211604148881299407159564, −9.856993838817028459367233246799, −8.760746773047692545320799849929, −7.68450973847761111023681387073, −6.08836786819580304643390178572, −4.91821747766078774047149817629, −3.05556374144475994004860024213, −1.14196292134310775067671760250, 0.45172628101231704806514469830, 1.44922318786353931893958605411, 4.88904828302003257623634326208, 6.18391013262816311297379030208, 6.88474872694296039142112499195, 7.65305452151072126340431801700, 8.727833940331947544000708824422, 10.17821432633369706685866686519, 10.77505920984269955131758489447, 11.64459358483245686670225629337

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