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Properties

Label 2-175-35.9-c3-0-17
Degree $2$
Conductor $175$
Sign $-0.669 + 0.742i$
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  + (−1.73 − i)2-s + (−6.06 + 3.5i)3-s + (−2.00 − 3.46i)4-s + 14·6-s + (12.1 + 14i)7-s + 24i·8-s + (11 − 19.0i)9-s + (2.5 + 4.33i)11-s + (24.2 + 13.9i)12-s + 14i·13-s + (−7 − 36.3i)14-s + (8.00 − 13.8i)16-s + (−18.1 + 10.5i)17-s + (−38.1 + 21.9i)18-s + (24.5 − 42.4i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.16 + 0.673i)3-s + (−0.250 − 0.433i)4-s + 0.952·6-s + (0.654 + 0.755i)7-s + 1.06i·8-s + (0.407 − 0.705i)9-s + (0.0685 + 0.118i)11-s + (0.583 + 0.336i)12-s + 0.298i·13-s + (−0.133 − 0.694i)14-s + (0.125 − 0.216i)16-s + (−0.259 + 0.149i)17-s + (−0.498 + 0.288i)18-s + (0.295 − 0.512i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.669 + 0.742i$
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: \((2,\ 175,\ (\ :3/2),\ -0.669 + 0.742i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0826525 - 0.185817i\)
\(L(\frac12)\) \(\approx\) \(0.0826525 - 0.185817i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-12.1 - 14i)T \)
good2 \( 1 + (1.73 + i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (6.06 - 3.5i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 14iT - 2.19e3T^{2} \)
17 \( 1 + (18.1 - 10.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-24.5 + 42.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (137. + 79.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (73.5 + 127. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (189. + 109.5i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 350T + 6.89e4T^{2} \)
43 \( 1 - 124iT - 7.95e4T^{2} \)
47 \( 1 + (454. + 262.5i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (262. - 151.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (52.5 + 90.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-206.5 + 357. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-359. + 207.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 432T + 3.57e5T^{2} \)
73 \( 1 + (-963. + 556.5i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (51.5 - 89.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 + (164.5 - 284. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 882iT - 9.12e5T^{2} \)
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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

−11.43332840597346799938067393139, −11.00662274561483517924828795489, −9.975228036499492926856476020329, −9.154613239194367076869781650206, −8.052450119012094090459107370437, −6.22032648252893880737690125290, −5.33198388048719196558540267537, −4.44089647609380881887977819851, −2.01119215369636447623047367429, −0.14931801372628997531005733865, 1.23967827047672135735462761624, 3.82842075125766111577848442654, 5.27722435464847098587106407376, 6.54670442800657062718079531739, 7.45697162630918669299296306914, 8.222491568389599267377798997111, 9.614626917995700643329239009835, 10.72048395195453522762142118797, 11.66008575319879216069525187806, 12.48463257589525170103262601361

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