↑

Properties

Label 2-425-5.4-c3-0-62
Degree $2$
Conductor $425$
Sign $-0.894 - 0.447i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3i·2-s + 8i·3-s − 4-s + 24·6-s − 28i·7-s − 21i·8-s − 37·9-s − 24·11-s − 8i·12-s + 58i·13-s − 84·14-s − 71·16-s + 17i·17-s + 111i·18-s − 116·19-s + ⋯
L(s)  = 1  − 1.06i·2-s + 1.53i·3-s − 0.125·4-s + 1.63·6-s − 1.51i·7-s − 0.928i·8-s − 1.37·9-s − 0.657·11-s − 0.192i·12-s + 1.23i·13-s − 1.60·14-s − 1.10·16-s + 0.242i·17-s + 1.45i·18-s − 1.40·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 - 17iT \)
good2 \( 1 + 3iT - 8T^{2} \)
3 \( 1 - 8iT - 27T^{2} \)
7 \( 1 + 28iT - 343T^{2} \)
11 \( 1 + 24T + 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
19 \( 1 + 116T + 6.85e3T^{2} \)
23 \( 1 - 60iT - 1.21e4T^{2} \)
29 \( 1 + 30T + 2.43e4T^{2} \)
31 \( 1 + 172T + 2.97e4T^{2} \)
37 \( 1 + 58iT - 5.06e4T^{2} \)
41 \( 1 + 342T + 6.89e4T^{2} \)
43 \( 1 - 148iT - 7.95e4T^{2} \)
47 \( 1 - 288iT - 1.03e5T^{2} \)
53 \( 1 + 318iT - 1.48e5T^{2} \)
59 \( 1 + 252T + 2.05e5T^{2} \)
61 \( 1 - 110T + 2.26e5T^{2} \)
67 \( 1 + 484iT - 3.00e5T^{2} \)
71 \( 1 + 708T + 3.57e5T^{2} \)
73 \( 1 + 362iT - 3.89e5T^{2} \)
79 \( 1 - 484T + 4.93e5T^{2} \)
83 \( 1 + 756iT - 5.71e5T^{2} \)
89 \( 1 - 774T + 7.04e5T^{2} \)
97 \( 1 + 382iT - 9.12e5T^{2} \)
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

−10.45003594599013903395813572461, −9.748953406935180400184091027290, −8.916466472894863703065734888805, −7.48714667875867588271705789486, −6.41697704808275461280133467235, −4.76542088483137001492988901804, −4.06550794064444127682719866466, −3.37760450585846030072698215390, −1.81424147318915866618309580227, 0, 2.00323208346037421144177330888, 2.73931685831830878603720113640, 5.21524494767023274350368859995, 5.84798190277569519804114226122, 6.61879848170905525825960160666, 7.52430748825825200745536874118, 8.317040883954338335083503122259, 8.772156140594064990213604724660, 10.48262812718079295191751081100

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