L-function 1-45-45.22-r1-0-0

• Label: 1-45-45.22-r1-0-0
• Degree: $1$
• Conductor: $45$
• Sign: $-0.665 + 0.746i$
• Analytic cond.: $4.83592$
• Root an. cond.: $4.83592$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 − 0.5i)2^-s + (0.5 + 0.866i)4^-s + (−0.866 − 0.5i)7^-s − i·8^-s + (−0.5 + 0.866i)11^-s + (−0.866 + 0.5i)13^-s + (0.5 + 0.866i)14^-s + (−0.5 + 0.866i)16^-s + i·17^-s − 19^-s + (0.866 − 0.5i)22^-s + (−0.866 + 0.5i)23^-s + 26^-s − i·28^-s + (0.5 − 0.866i)29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(45\)    =    \(3^{2} \cdot 5\)
Sign:	$-0.665 + 0.746i$
Analytic conductor:	\(4.83592\)
Root analytic conductor:	\(4.83592\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{45} (22, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 45,\ (1:\ ),\ -0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.07845651932 + 0.1750534138i\)
\(L(1)\)	\(\approx\)	\(0.4840460907 + 0.02776180649i\)

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−34.06151056700494650772702719454, −32.465412585242615457893248284591, −31.75453362066903434375373047403, −29.66562785068075368286224428318, −28.93093951479350805661633250416, −27.64334081683377480495073950317, −26.59585467487252142232467659889, −25.47762892432451591033497171155, −24.51793521034089750397067975770, −23.206262398087601627528370319292, −21.7606620514005981723005607636, −20.07753674170606006718329670245, −19.04151011940460404357621385339, −17.998963101725770418656010512276, −16.521154234023988325928725958556, −15.681933987404926295541551153470, −14.25403148682456445864193835904, −12.48733080499591676669872992166, −10.78182061488903454498294851264, −9.53994252934195705096009894911, −8.27261626325058397475706752731, −6.74324671982229943913575667767, −5.39857089607537723104592654746, −2.670974370481482485253676789946, −0.14130967503481220922695250696, 2.19101918745459801575242198925, 4.032044497248041886005334707141, 6.60124187534404225208310724396, 7.90780778592658269364481350704, 9.58487706885013852619696386391, 10.436159584163931815994561229096, 12.105522055852805717440229925110, 13.18032560715474647607934810090, 15.22739345213525343768046309446, 16.632214765119503901068312226427, 17.57245308434466399562898595143, 19.083769072238352475164183530116, 19.87078748891966752377397858965, 21.175880296566698468686602201851, 22.427332458474392127167218756659, 23.94343793621385581750943992136, 25.652203151772568378219369107323, 26.20907046014930731722848414309, 27.58115119414771634982704375590, 28.69579349803326539059431930066, 29.584584025337479586650778906892, 30.74172481665584982337739576228, 32.07519662578403889431059635267, 33.597219127631391998114051153161, 34.655451582689289578842579421954

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