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Properties

Label 2-175-7.2-c3-0-10
Degree $2$
Conductor $175$
Sign $0.266 - 0.963i$
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 − 1.73i)2-s + (3.5 + 6.06i)3-s + (2.00 + 3.46i)4-s + 14·6-s + (−14 + 12.1i)7-s + 24·8-s + (−11 + 19.0i)9-s + (2.5 + 4.33i)11-s + (−13.9 + 24.2i)12-s + 14·13-s + (7 + 36.3i)14-s + (8.00 − 13.8i)16-s + (−10.5 − 18.1i)17-s + (21.9 + 38.1i)18-s + (−24.5 + 42.4i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.673 + 1.16i)3-s + (0.250 + 0.433i)4-s + 0.952·6-s + (−0.755 + 0.654i)7-s + 1.06·8-s + (−0.407 + 0.705i)9-s + (0.0685 + 0.118i)11-s + (−0.336 + 0.583i)12-s + 0.298·13-s + (0.133 + 0.694i)14-s + (0.125 − 0.216i)16-s + (−0.149 − 0.259i)17-s + (0.288 + 0.498i)18-s + (−0.295 + 0.512i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.06350 + 1.56982i\)
\(L(\frac12)\) \(\approx\) \(2.06350 + 1.56982i\)
\(L(\frac{5}{2})\) 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 \( 1 + (14 - 12.1i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-3.5 - 6.06i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 14T + 2.19e3T^{2} \)
17 \( 1 + (10.5 + 18.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (24.5 - 42.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (79.5 - 137. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 58T + 2.43e4T^{2} \)
31 \( 1 + (73.5 + 127. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-109.5 + 189. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 350T + 6.89e4T^{2} \)
43 \( 1 - 124T + 7.95e4T^{2} \)
47 \( 1 + (-262.5 + 454. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-151.5 - 262. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
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 \( 1 + (-207.5 - 359. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 432T + 3.57e5T^{2} \)
73 \( 1 + (556.5 + 963. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-51.5 + 89.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (-164.5 + 284. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 882T + 9.12e5T^{2} \)
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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

−12.41518753836637305778339618675, −11.49147566099448896290467617327, −10.41540815402615826981706246559, −9.568486452105956499056169967799, −8.682444010550874964209279146861, −7.43130851241880167022147266125, −5.82438280626747790386050284338, −4.23946115791873206826134297363, −3.47116203979427545720376204712, −2.34101773116013074874161667522, 1.03233972621647082205931848542, 2.56634176846614565735567450493, 4.32820213747818148535065719343, 6.07529306323118492540058579997, 6.77299569867146247650764950554, 7.59317101008746325687358176120, 8.674095172644260194916897755843, 10.08049444757254292743778817189, 11.01657904965880234590145586554, 12.53314329666441598999732052709

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