L-function 1-664-664.165-r1-0-0

• Label: 1-664-664.165-r1-0-0
• Degree: $1$
• Conductor: $664$
• Sign: $1$
• Analytic cond.: $71.3567$
• Root an. cond.: $71.3567$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 3^-s + 5^-s + 7^-s + 9^-s − 11^-s + 13^-s − 15^-s + 17^-s + 19^-s − 21^-s + 23^-s + 25^-s − 27^-s − 29^-s + 31^-s + 33^-s + 35^-s − 37^-s − 39^-s + 41^-s + 43^-s + 45^-s − 47^-s + 49^-s − 51^-s + 53^-s − 55^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$664$    =    $2^{3} \cdot 83$
Sign:	$1$
Analytic conductor:	$71.3567$
Root analytic conductor:	$71.3567$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	$\chi_{664} (165, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(1,\ 664,\ (1:\ ),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.426574226$
$L(1)$	$\approx$	$1.219174394$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	83	$1$
good	3	$1 - T$
	5	$1 + T$
	7	$1 + T$
	11	$1 - T$
	13	$1 + T$
	17	$1 + T$
	19	$1 + T$
	23	$1 + T$
	29	$1 - T$

Imaginary part of the first few zeros on the critical line

−22.7740995421901859192081173228, −21.62679627194695333443342961583, −20.936196562259149381446212084, −20.74151289466099771480197293994, −18.90273586087853229047979049470, −18.29012396191195618004372684357, −17.72401033203860753224937399579, −16.96895508981321198987612945097, −16.13535823918146349277061252392, −15.253576697845536504656788894282, −14.14181886733345952946389848413, −13.38429032723916218707511015021, −12.56926844991406536388321506934, −11.51712323384679781054405406486, −10.797592306226484300697869103588, −10.1355262777652321752784913734, −9.1081758417829043482873129001, −7.916504638350049121326854193730, −7.06004599684196330146172488494, −5.75944615737523917020031999341, −5.462077450735719058500255357835, −4.50921627787091457746011965130, −3.02880733763852492788508882948, −1.65096963303009147087346008242, −0.91983156285504112405518249016, 0.91983156285504112405518249016, 1.65096963303009147087346008242, 3.02880733763852492788508882948, 4.50921627787091457746011965130, 5.462077450735719058500255357835, 5.75944615737523917020031999341, 7.06004599684196330146172488494, 7.916504638350049121326854193730, 9.1081758417829043482873129001, 10.1355262777652321752784913734, 10.797592306226484300697869103588, 11.51712323384679781054405406486, 12.56926844991406536388321506934, 13.38429032723916218707511015021, 14.14181886733345952946389848413, 15.253576697845536504656788894282, 16.13535823918146349277061252392, 16.96895508981321198987612945097, 17.72401033203860753224937399579, 18.29012396191195618004372684357, 18.90273586087853229047979049470, 20.74151289466099771480197293994, 20.936196562259149381446212084, 21.62679627194695333443342961583, 22.7740995421901859192081173228

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