L-function 1-2340-2340.2167-r0-0-0

• Label: 1-2340-2340.2167-r0-0-0
• Degree: $1$
• Conductor: $2340$
• Sign: $-0.401 - 0.916i$
• Analytic cond.: $10.8669$
• Root an. cond.: $10.8669$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− i·7^-s + (0.5 − 0.866i)11^-s + (−0.866 − 0.5i)17^-s + (−0.5 + 0.866i)19^-s − i·23^-s + (0.5 − 0.866i)29^-s + (0.5 − 0.866i)31^-s + (0.866 − 0.5i)37^-s + 41^-s − i·43^-s + (−0.866 + 0.5i)47^-s − 49^-s − i·53^-s + (−0.5 − 0.866i)59^-s + 61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign:	$-0.401 - 0.916i$
Analytic conductor:	\(10.8669\)
Root analytic conductor:	\(10.8669\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2340} (2167, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 2340,\ (0:\ ),\ -0.401 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.6935212963 - 1.060819489i\)
\(L(1)\)	\(\approx\)	\(0.9726391243 - 0.2813847522i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	3	\( 1 \)
	5	\( 1 \)
	13	\( 1 \)
good	7	\( 1 - iT \)
	11	\( 1 + (0.5 - 0.866i)T \)
	17	\( 1 + (-0.866 - 0.5i)T \)
	19	\( 1 + (-0.5 + 0.866i)T \)
	23	\( 1 - iT \)
	29	\( 1 + (0.5 - 0.866i)T \)
	31	\( 1 + (0.5 - 0.866i)T \)
	37	\( 1 + (0.866 - 0.5i)T \)
	41	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−19.84169148933669478795122128984, −19.15111697339367239312503450955, −18.307446265113392726778440699308, −17.73007532748916578509026884290, −17.10361910964917438415596258892, −16.12721926683396948826576420334, −15.45412006368216540390295800786, −14.86624780657436238927993337718, −14.2397442453636731782011330495, −13.14047645649362166582121141062, −12.57471652314377884898388418869, −11.94608355404777837748680736740, −11.11432116101866669812448652601, −10.35761642131149146418718608040, −9.457987720925199137720619324, −8.75304973629533571067040406646, −8.275511446045208398609222230341, −6.94078134902031856140780863897, −6.601238881391816346935750988340, −5.60317799474330435981436004597, −4.69945156462521727823957920903, −4.12866664287937178766657984303, −2.79915763504881418148635941179, −2.28054730285173469699517869495, −1.2259548391172804330518495255, 0.4361690499774632750995400912, 1.41049708534261004211479316915, 2.49002117031462093649055013544, 3.55302499116708064226528719597, 4.13818062734382896515128198687, 4.98169002610660634322202900512, 6.19556922158126014819237011637, 6.49596068073080299655471809415, 7.74448453939302340538951941272, 8.04868236829148815504273331713, 9.26300728890723350847870535613, 9.73618283075653258906985136620, 10.7827212074291686217914352870, 11.256437592488127020140542974639, 11.9980756724637956375285178463, 13.14108119321628122083172139735, 13.507393640761315821661936722421, 14.30324659458326994004770583353, 14.940138822565316837525644880703, 16.06259500124085777917846323364, 16.37964105827472148981183705372, 17.35413422143143051693270199334, 17.7129641622349332109422297985, 18.80288834805779489846305079447, 19.45243740868814249851735189850

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