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Properties

Label 2-175-35.9-c3-0-24
Degree $2$
Conductor $175$
Sign $0.315 + 0.949i$
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  + (3.33 + 1.92i)2-s + (−7.76 + 4.48i)3-s + (3.39 + 5.88i)4-s − 34.4·6-s + (−13.8 − 12.3i)7-s − 4.64i·8-s + (26.7 − 46.2i)9-s + (−11.5 − 20.0i)11-s + (−52.7 − 30.4i)12-s + 61.0i·13-s + (−22.2 − 67.6i)14-s + (36.0 − 62.5i)16-s + (−0.596 + 0.344i)17-s + (177. − 102. i)18-s + (31.6 − 54.7i)19-s + ⋯
L(s)  = 1  + (1.17 + 0.679i)2-s + (−1.49 + 0.862i)3-s + (0.424 + 0.735i)4-s − 2.34·6-s + (−0.745 − 0.666i)7-s − 0.205i·8-s + (0.989 − 1.71i)9-s + (−0.316 − 0.548i)11-s + (−1.26 − 0.732i)12-s + 1.30i·13-s + (−0.424 − 1.29i)14-s + (0.564 − 0.976i)16-s + (−0.00850 + 0.00490i)17-s + (2.33 − 1.34i)18-s + (0.381 − 0.661i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.315 + 0.949i$
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: \((2,\ 175,\ (\ :3/2),\ 0.315 + 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.493707 - 0.356276i\)
\(L(\frac12)\) \(\approx\) \(0.493707 - 0.356276i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.8 + 12.3i)T \)
good2 \( 1 + (-3.33 - 1.92i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (7.76 - 4.48i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (11.5 + 20.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 61.0iT - 2.19e3T^{2} \)
17 \( 1 + (0.596 - 0.344i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-31.6 + 54.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (107. + 62.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 104.T + 2.43e4T^{2} \)
31 \( 1 + (140. + 242. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (228. + 131. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 243.T + 6.89e4T^{2} \)
43 \( 1 - 172. iT - 7.95e4T^{2} \)
47 \( 1 + (92.9 + 53.6i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-38.8 + 22.4i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-228. - 395. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (236. - 410. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-198. + 114. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 407.T + 3.57e5T^{2} \)
73 \( 1 + (301. - 174. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-420. + 727. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 885. iT - 5.71e5T^{2} \)
89 \( 1 + (428. - 741. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 189. iT - 9.12e5T^{2} \)
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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

−12.08506232212818506846552784201, −11.25479256776260257272099569153, −10.22699147985012804248171157464, −9.353804597450380970562297371379, −7.19367404491147584129357330361, −6.35378499624710292857260192267, −5.57777706946123597598108465617, −4.47381934898725499859787078388, −3.72049546492953291748846848905, −0.21964600931817025925112850350, 1.82260437262016184238413900765, 3.41082310202111241785301488519, 5.24640452484196390400340848663, 5.58795005167364177740673920216, 6.77112437841602536748483923984, 8.073949786389934926122310282635, 10.02822005102080991770776611143, 10.87735937397874704627128389662, 11.93951237181382044145814718385, 12.42713970745056358240839113296

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