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Properties

Label 2-175-35.9-c3-0-32
Degree $2$
Conductor $175$
Sign $0.192 + 0.981i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.192 + 0.981i$
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: \((2,\ 175,\ (\ :3/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.18710 - 1.80003i\)
\(L(\frac12)\) \(\approx\) \(2.18710 - 1.80003i\)
\(L(\frac{5}{2})\) 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

−12.69284770322988219734524032451, −11.00940449351453038228169306905, −9.743980282060787498086468623528, −9.060247483031425653912119698547, −7.59056145309397360110394507302, −6.95161059913089782740263716462, −5.61748806459738988525750806906, −4.11671388402718838197787131014, −2.92883879202137655784970040231, −1.00320797985453228838312600931, 2.52857199056945698979615513243, 3.35642694053906931968702762451, 4.39001246857981053684918402440, 5.77927391683633904544417129589, 7.52741563476860634347429557876, 8.824383975292032683366638587331, 9.096470481821005127502885217045, 10.40775216757844745946815289102, 11.72504226598861043253689173962, 12.63794663721261870191020588007

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