↑

Properties

Label 2-425-85.9-c1-0-17
Degree $2$
Conductor $425$
Sign $0.341 - 0.939i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 + 1.27i)2-s + (−0.263 + 0.635i)3-s + 1.26i·4-s + (−1.14 + 0.475i)6-s + (4.01 − 1.66i)7-s + (0.943 − 0.943i)8-s + (1.78 + 1.78i)9-s + (0.0485 − 0.0200i)11-s + (−0.801 − 0.331i)12-s − 3.02·13-s + (7.24 + 3.00i)14-s + 4.93·16-s + (−2.69 − 3.12i)17-s + 4.56i·18-s + (−5.52 + 5.52i)19-s + ⋯
L(s)  = 1  + (0.902 + 0.902i)2-s + (−0.151 + 0.366i)3-s + 0.630i·4-s + (−0.468 + 0.194i)6-s + (1.51 − 0.628i)7-s + (0.333 − 0.333i)8-s + (0.595 + 0.595i)9-s + (0.0146 − 0.00605i)11-s + (−0.231 − 0.0958i)12-s − 0.839·13-s + (1.93 + 0.801i)14-s + 1.23·16-s + (−0.653 − 0.757i)17-s + 1.07i·18-s + (−1.26 + 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.341 - 0.939i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.341 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97558 + 1.38365i\)
\(L(\frac12)\) \(\approx\) \(1.97558 + 1.38365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (2.69 + 3.12i)T \)
good2 \( 1 + (-1.27 - 1.27i)T + 2iT^{2} \)
3 \( 1 + (0.263 - 0.635i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-4.01 + 1.66i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.0485 + 0.0200i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
19 \( 1 + (5.52 - 5.52i)T - 19iT^{2} \)
23 \( 1 + (-0.398 - 0.962i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-0.161 + 0.388i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.27 + 0.529i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.128 + 0.311i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.52 - 6.09i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-7.06 + 7.06i)T - 43iT^{2} \)
47 \( 1 + 6.13T + 47T^{2} \)
53 \( 1 + (8.52 + 8.52i)T + 53iT^{2} \)
59 \( 1 + (3.60 + 3.60i)T + 59iT^{2} \)
61 \( 1 + (2.28 + 5.51i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 0.916iT - 67T^{2} \)
71 \( 1 + (-3.86 - 1.59i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.98 - 2.06i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (9.22 - 3.82i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (-17.7 - 7.35i)T + (68.5 + 68.5i)T^{2} \)
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

−11.24779316100652284851835457159, −10.58097968434311585377909795853, −9.694298367022683273028630620663, −8.126127665154838174701592474612, −7.57300428408066259622387326174, −6.63400901649112944103374065453, −5.33275541612809914949535307783, −4.67680681040740641512744179710, −4.04575456639533287008351760629, −1.83161054985129126826578386517, 1.66468095722361214288446201574, 2.58441005616308547281335952857, 4.28693693841260205117080478426, 4.76420723231279769150005610593, 6.01273526944170021520686706285, 7.27308457825965169535956132268, 8.260432715565669215948134730360, 9.221654833617291563773007674819, 10.66052319710973270260214921501, 11.15642568821090710014202652684

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