L-function 2-350-5.4-c1-0-3

• Label: 2-350-5.4-c1-0-3
• Degree: $2$
• Conductor: $350$
• Sign: $0.894 - 0.447i$
• Analytic cond.: $2.79476$
• Root an. cond.: $1.67175$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·2^-s − i·3^-s − 4^-s + 6^-s + i·7^-s − i·8^-s + 2·9^-s + 3·11^-s + i·12^-s − 2i·13^-s − 14^-s + 16^-s + 3i·17^-s + 2i·18^-s + 7·19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$350$    =    $2 \cdot 5^{2} \cdot 7$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$2.79476$
Root analytic conductor:	$1.67175$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{350} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 350,\ (\ :1/2),\ 0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.36870 + 0.323107i$
$L(\frac{3}{2})$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$1 - iT$
	5	$1$
	7	$1 - iT$
good	3	$1 + iT - 3T^{2}$
	11	$1 - 3T + 11T^{2}$
	13	$1 + 2iT - 13T^{2}$
	17	$1 - 3iT - 17T^{2}$
	19	$1 - 7T + 19T^{2}$
	23	$1 - 23T^{2}$
	29	$1 - 6T + 29T^{2}$
	31	$1 + 4T + 31T^{2}$
	37	$1 - 8iT - 37T^{2}$

Imaginary part of the first few zeros on the critical line

−11.92881269295989137345621351552, −10.47158512922332734982283242374, −9.568685522922711961902715967641, −8.585608861462356437225398783181, −7.64782974353942160188456327130, −6.81224159074683869205437025071, −5.92173721221202844095837243996, −4.75532718895897824441470639549, −3.38090030408863777519563102748, −1.39931523367926616697580766515, 1.38643737698553398363216844504, 3.22476268795815471130630124211, 4.22140969165292187868312288693, 5.12974229193952936245223353764, 6.67972868613410665024264603752, 7.66578944834140103048419730306, 9.137854964217652677403368906993, 9.579009428886797063318045648029, 10.47171892529940308758171697404, 11.43542080794687841361161320888

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