L-function 2-425-5.4-c1-0-13

• Label: 2-425-5.4-c1-0-13
• Degree: $2$
• Conductor: $425$
• Sign: $0.894 - 0.447i$
• Analytic cond.: $3.39364$
• Root an. cond.: $1.84218$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·2^-s + 4^-s − 4i·7^-s + 3i·8^-s + 3·9^-s − 2i·13^-s + 4·14^-s − 16^-s − i·17^-s + 3i·18^-s + 4·19^-s + 4i·23^-s + 2·26^-s − 4i·28^-s − 6·29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$425$    =    $5^{2} \cdot 17$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$3.39364$
Root analytic conductor:	$1.84218$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{425} (324, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 425,\ (\ :1/2),\ 0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.69010 + 0.398979i$
$L(\frac{3}{2})$		not available

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	17	$1 + iT$
good	2	$1 - iT - 2T^{2}$
	3	$1 - 3T^{2}$
	7	$1 + 4iT - 7T^{2}$
	11	$1 + 11T^{2}$
	13	$1 + 2iT - 13T^{2}$
	19	$1 - 4T + 19T^{2}$
	23	$1 - 4iT - 23T^{2}$
	29	$1 + 6T + 29T^{2}$
	31	$1 - 4T + 31T^{2}$

Imaginary part of the first few zeros on the critical line

−11.12545955184019786026691703065, −10.34419994153612588694837079470, −9.580103581843853268273104393610, −8.054755129932527360045391104644, −7.37323907538424822017392644714, −6.88283093269140417083303385877, −5.64558969832405997965298070006, −4.50562702493167937967098019859, −3.27421846633199865520241686865, −1.39699613843806692195408059085, 1.66415178220154260649314063962, 2.67920169036367087688039689822, 3.96408590383967063163987482962, 5.34179206358898239150626680022, 6.43689548753242383069829388258, 7.30700044211496192342148109860, 8.571556531784596602191283720981, 9.526145800625271601965973157201, 10.20379975024811324405866717620, 11.30343291001595261223671925186

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