L-function 1-520-520.259-r1-0-0

• Label: 1-520-520.259-r1-0-0
• Degree: $1$
• Conductor: $520$
• Sign: $1$
• Analytic cond.: $55.8817$
• Root an. cond.: $55.8817$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 3^-s − 7^-s + 9^-s − 11^-s − 17^-s − 19^-s + 21^-s + 23^-s − 27^-s − 29^-s + 31^-s + 33^-s − 37^-s − 41^-s − 43^-s − 47^-s + 49^-s + 51^-s + 53^-s + 57^-s − 59^-s − 61^-s − 63^-s + 67^-s − 69^-s + 71^-s + 73^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$520$    =    $2^{3} \cdot 5 \cdot 13$
Sign:	$1$
Analytic conductor:	$55.8817$
Root analytic conductor:	$55.8817$
Motivic weight:	$0$
Rational:	yes
Arithmetic:	yes
Character:	$\chi_{520} (259, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(1,\ 520,\ (1:\ ),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.5148202990$
$L(1)$	$\approx$	$0.5510718060$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	13	$1$
good	3	$1 - T$
	7	$1 - T$
	11	$1 - T$
	17	$1 - T$
	19	$1 - T$
	23	$1 + T$
	29	$1 - T$
	31	$1 + T$
	37	$1 - T$

Imaginary part of the first few zeros on the critical line

−23.09707377972868128018937781483, −22.72418238364784555759794173295, −21.70491731815742168912800708006, −21.0482588132289447096488611110, −19.88298558957835339677332766670, −18.907317891868830967075490635568, −18.33069348129050378369843404392, −17.222503621981959000857993914435, −16.67709708897373370975893943168, −15.61706518541201909087792985322, −15.22090808881279009348537786166, −13.45200329371061067476258440370, −13.01908459021767277876184131691, −12.14952871468906681125493356351, −11.04508347773828008508857891623, −10.41596331304031995801954993560, −9.511526498722929217846882569523, −8.36237745467781401456136119648, −7.02788207198529074584542600041, −6.46796012170503251356478368880, −5.409260178176494348875078319853, −4.53173229427681136337963490431, −3.29460287980216002325118225496, −1.99999002129155580385294339465, −0.38464189291356244580902717207, 0.38464189291356244580902717207, 1.99999002129155580385294339465, 3.29460287980216002325118225496, 4.53173229427681136337963490431, 5.409260178176494348875078319853, 6.46796012170503251356478368880, 7.02788207198529074584542600041, 8.36237745467781401456136119648, 9.511526498722929217846882569523, 10.41596331304031995801954993560, 11.04508347773828008508857891623, 12.14952871468906681125493356351, 13.01908459021767277876184131691, 13.45200329371061067476258440370, 15.22090808881279009348537786166, 15.61706518541201909087792985322, 16.67709708897373370975893943168, 17.222503621981959000857993914435, 18.33069348129050378369843404392, 18.907317891868830967075490635568, 19.88298558957835339677332766670, 21.0482588132289447096488611110, 21.70491731815742168912800708006, 22.72418238364784555759794173295, 23.09707377972868128018937781483

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