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Properties

Label 2-175-5.4-c9-0-80
Degree $2$
Conductor $175$
Sign $0.894 - 0.447i$
Analytic cond. $90.1312$
Root an. cond. $9.49374$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28i·2-s − 116i·3-s − 272·4-s − 3.24e3·6-s − 2.40e3i·7-s − 6.72e3i·8-s + 6.22e3·9-s − 2.55e4·11-s + 3.15e4i·12-s − 4.23e4i·13-s − 6.72e4·14-s − 3.27e5·16-s + 5.26e5i·17-s − 1.74e5i·18-s + 3.50e5·19-s + ⋯
L(s)  = 1  − 1.23i·2-s − 0.826i·3-s − 0.531·4-s − 1.02·6-s − 0.377i·7-s − 0.580i·8-s + 0.316·9-s − 0.526·11-s + 0.439i·12-s − 0.410i·13-s − 0.467·14-s − 1.24·16-s + 1.52i·17-s − 0.391i·18-s + 0.616·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(90.1312\)
Root analytic conductor: \(9.49374\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :9/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.06797680924\)
\(L(\frac12)\) \(\approx\) \(0.06797680924\)
\(L(\frac{11}{2})\) 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 \( 1 + 2.40e3iT \)
good2 \( 1 + 28iT - 512T^{2} \)
3 \( 1 + 116iT - 1.96e4T^{2} \)
11 \( 1 + 2.55e4T + 2.35e9T^{2} \)
13 \( 1 + 4.23e4iT - 1.06e10T^{2} \)
17 \( 1 - 5.26e5iT - 1.18e11T^{2} \)
19 \( 1 - 3.50e5T + 3.22e11T^{2} \)
23 \( 1 + 6.21e5iT - 1.80e12T^{2} \)
29 \( 1 + 6.72e6T + 1.45e13T^{2} \)
31 \( 1 + 6.41e6T + 2.64e13T^{2} \)
37 \( 1 - 2.31e6iT - 1.29e14T^{2} \)
41 \( 1 + 1.02e7T + 3.27e14T^{2} \)
43 \( 1 - 3.01e7iT - 5.02e14T^{2} \)
47 \( 1 - 2.36e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.72e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.49e7T + 8.66e15T^{2} \)
61 \( 1 - 1.46e7T + 1.16e16T^{2} \)
67 \( 1 - 2.44e8iT - 2.72e16T^{2} \)
71 \( 1 - 6.19e7T + 4.58e16T^{2} \)
73 \( 1 + 2.83e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.76e8T + 1.19e17T^{2} \)
83 \( 1 + 7.29e7iT - 1.86e17T^{2} \)
89 \( 1 - 8.96e8T + 3.50e17T^{2} \)
97 \( 1 + 1.20e9iT - 7.60e17T^{2} \)
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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

−10.41303416479171704547848240963, −9.416358635380412097278021659071, −7.971298553194077500011485821295, −7.12309892097355287467949277125, −5.91664450131519930752339822026, −4.27160892903382033168473247630, −3.20432161263086154664020240966, −1.93316565915546368717561474640, −1.23242893207099596914871646219, −0.01446230521967327860851920175, 2.04611581031401876354428720246, 3.57546077147594006242642379311, 5.00703186109405715450804645380, 5.49651839571519605956052754671, 6.98041080334226520259682781715, 7.61037842284467470986765016175, 8.985528234511624577613196419023, 9.604424521197358853941123539827, 10.91698949898015833804265695104, 11.77818916129660758820564015951

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