↑

Properties

Label 2-175-5.4-c9-0-4
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 41.8i·2-s − 0.232i·3-s − 1.23e3·4-s + 9.71·6-s + 2.40e3i·7-s − 3.02e4i·8-s + 1.96e4·9-s + 1.74e4·11-s + 287. i·12-s + 1.22e5i·13-s − 1.00e5·14-s + 6.31e5·16-s + 3.31e5i·17-s + 8.22e5i·18-s − 7.61e5·19-s + ⋯
L(s)  = 1  + 1.84i·2-s − 0.00165i·3-s − 2.41·4-s + 0.00305·6-s + 0.377i·7-s − 2.61i·8-s + 0.999·9-s + 0.358·11-s + 0.00399i·12-s + 1.18i·13-s − 0.698·14-s + 2.40·16-s + 0.963i·17-s + 1.84i·18-s − 1.34·19-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

\(L(5)\) \(\approx\) \(0.2349859787\)
\(L(\frac12)\) \(\approx\) \(0.2349859787\)
\(L(\frac{11}{2})\) 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 \( 1 - 41.8iT - 512T^{2} \)
3 \( 1 + 0.232iT - 1.96e4T^{2} \)
11 \( 1 - 1.74e4T + 2.35e9T^{2} \)
13 \( 1 - 1.22e5iT - 1.06e10T^{2} \)
17 \( 1 - 3.31e5iT - 1.18e11T^{2} \)
19 \( 1 + 7.61e5T + 3.22e11T^{2} \)
23 \( 1 + 1.23e6iT - 1.80e12T^{2} \)
29 \( 1 + 6.34e5T + 1.45e13T^{2} \)
31 \( 1 + 5.38e6T + 2.64e13T^{2} \)
37 \( 1 + 3.03e6iT - 1.29e14T^{2} \)
41 \( 1 + 7.37e6T + 3.27e14T^{2} \)
43 \( 1 - 2.06e7iT - 5.02e14T^{2} \)
47 \( 1 - 2.03e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.97e7iT - 3.29e15T^{2} \)
59 \( 1 + 6.03e7T + 8.66e15T^{2} \)
61 \( 1 + 9.44e6T + 1.16e16T^{2} \)
67 \( 1 + 2.19e8iT - 2.72e16T^{2} \)
71 \( 1 + 5.58e7T + 4.58e16T^{2} \)
73 \( 1 + 4.54e8iT - 5.88e16T^{2} \)
79 \( 1 + 4.51e7T + 1.19e17T^{2} \)
83 \( 1 - 3.34e8iT - 1.86e17T^{2} \)
89 \( 1 + 6.51e8T + 3.50e17T^{2} \)
97 \( 1 + 1.42e9iT - 7.60e17T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.39771453510329306207291881941, −10.70113266286014843337724186097, −9.420467794446774905005740129243, −8.747124952924361580484625026152, −7.71117019110548800070389249248, −6.66319043224519268837625477050, −6.12480597037519582849115107964, −4.67009652998445790268482453126, −4.02113796465432456286662043056, −1.69670538405870065630962363110, 0.05888862680510438162117179853, 1.07632412304607254020876303177, 2.09522946859034318198371959893, 3.36463948561674419248831347104, 4.21189683884799826338780450399, 5.32392643963797378463785601067, 7.16254627019014735080741584377, 8.476867938468975090877531533661, 9.574122841036729857225694060983, 10.26938908899253293883309211308

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