Properties

Label 2-175-7.6-c2-0-22
Degree $2$
Conductor $175$
Sign $-1$
Analytic cond. $4.76840$
Root an. cond. $2.18366$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4.47i·3-s − 3·4-s − 4.47i·6-s − 7·7-s − 7·8-s − 11.0·9-s + 2·11-s + 13.4i·12-s + 13.4i·13-s − 7·14-s + 5·16-s − 26.8i·17-s − 11.0·18-s − 13.4i·19-s + ⋯
L(s)  = 1  + 0.5·2-s − 1.49i·3-s − 0.750·4-s − 0.745i·6-s − 7-s − 0.875·8-s − 1.22·9-s + 0.181·11-s + 1.11i·12-s + 1.03i·13-s − 0.5·14-s + 0.312·16-s − 1.57i·17-s − 0.611·18-s − 0.706i·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(4.76840\)
Root analytic conductor: \(2.18366\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(-0.783303i\)
\(L(\frac12)\) \(\approx\) \(-0.783303i\)
\(L(2)\) 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 \( 1 + 7T \)
good2 \( 1 - T + 4T^{2} \)
3 \( 1 + 4.47iT - 9T^{2} \)
11 \( 1 - 2T + 121T^{2} \)
13 \( 1 - 13.4iT - 169T^{2} \)
17 \( 1 + 26.8iT - 289T^{2} \)
19 \( 1 + 13.4iT - 361T^{2} \)
23 \( 1 + 26T + 529T^{2} \)
29 \( 1 + 22T + 841T^{2} \)
31 \( 1 + 53.6iT - 961T^{2} \)
37 \( 1 + 14T + 1.36e3T^{2} \)
41 \( 1 + 26.8iT - 1.68e3T^{2} \)
43 \( 1 - 34T + 1.84e3T^{2} \)
47 \( 1 - 26.8iT - 2.20e3T^{2} \)
53 \( 1 - 34T + 2.80e3T^{2} \)
59 \( 1 + 40.2iT - 3.48e3T^{2} \)
61 \( 1 - 93.9iT - 3.72e3T^{2} \)
67 \( 1 + 14T + 4.48e3T^{2} \)
71 \( 1 - 62T + 5.04e3T^{2} \)
73 \( 1 - 53.6iT - 5.32e3T^{2} \)
79 \( 1 - 38T + 6.24e3T^{2} \)
83 \( 1 + 40.2iT - 6.88e3T^{2} \)
89 \( 1 + 26.8iT - 7.92e3T^{2} \)
97 \( 1 - 26.8iT - 9.40e3T^{2} \)
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

−12.20647871316742965032368694329, −11.56866256009431629392314537405, −9.670426737687766323550471917527, −8.985239900909979527651130182768, −7.57536351186365520859072305526, −6.67164245295287379208520995705, −5.70072161390158234500649530094, −4.09769343200612127736577170691, −2.50371016408415745461142317959, −0.40019394589696948159491333001, 3.40473490249149856042492731357, 3.95647343698359600242866392631, 5.29010600518441835995867457971, 6.15150899112071731328165737041, 8.208358749615457412753892010061, 9.165939351648076865556463342584, 10.08304732460621985622511326694, 10.57461699136790237582817836570, 12.25423672387255236361291503470, 12.91536394992629023881575693461

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