L-function 1-13e2-169.105-r0-0-0

• Label: 1-13e2-169.105-r0-0-0
• Degree: $1$
• Conductor: $169$
• Sign: $-0.986 - 0.166i$
• Analytic cond.: $0.784832$
• Root an. cond.: $0.784832$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.970 − 0.239i)2^-s + (0.120 − 0.992i)3^-s + (0.885 + 0.464i)4^-s + (0.568 − 0.822i)5^-s + (−0.354 + 0.935i)6^-s + (−0.748 − 0.663i)7^-s + (−0.748 − 0.663i)8^-s + (−0.970 − 0.239i)9^-s + (−0.748 + 0.663i)10^-s + (−0.970 + 0.239i)11^-s + (0.568 − 0.822i)12^-s + (0.568 + 0.822i)14^-s + (−0.748 − 0.663i)15^-s + (0.568 + 0.822i)16^-s + (−0.748 − 0.663i)17^-s + (0.885 + 0.464i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$169$    =    $13^{2}$
Sign:	$-0.986 - 0.166i$
Analytic conductor:	$0.784832$
Root analytic conductor:	$0.784832$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{169} (105, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 169,\ (0:\ ),\ -0.986 - 0.166i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.04684593783 - 0.5587037508i$
$L(1)$	$\approx$	$0.4702348160 - 0.4270755752i$

Euler product

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

	$p$	$F_p(T)$
bad	13	$1$
good	2	$1 + (-0.970 - 0.239i)T$
	3	$1 + (0.120 - 0.992i)T$
	5	$1 + (0.568 - 0.822i)T$
	7	$1 + (-0.748 - 0.663i)T$
	11	$1 + (-0.970 + 0.239i)T$
	17	$1 + (-0.748 - 0.663i)T$
	19	$1 + T$
	23	$1 + T$
	29	$1 + (-0.970 - 0.239i)T$

Imaginary part of the first few zeros on the critical line

−28.064831859686738777672311711212, −26.69528861256239627619664795827, −26.32944174162826013340466900737, −25.57103669131064739253753074685, −24.600155009466140085740486900446, −23.07958709884468864107060847599, −22.05198913230482357796891779303, −21.247599330928753591492376899718, −20.19366846685784293807367325439, −19.06149575248825848228952861621, −18.28264049966145027969603095033, −17.21616344165260690503979146276, −16.12906899433927437376025177517, −15.385100331120555527668846407467, −14.606806788161078621222603722264, −13.14644359289650280898793749713, −11.3427789151020566877961680325, −10.62244025967653013410354913356, −9.64519269260633574298597106665, −8.97974611586246229658617917448, −7.58619488591752458821879481497, −6.19529016767813713632753856513, −5.39786486951356377877627403091, −3.2310370062479135462161270296, −2.36537918907374508224377269129, 0.58132552610903325239765974375, 1.93805835945170957887258274566, 3.169346791446382646398921753743, 5.35071787479851751212790773928, 6.75421823972400754009483827970, 7.52638027604034043047626865747, 8.75209945467928473478047677319, 9.60852069832059301455936691684, 10.80144980223519908247453958942, 12.11958108422110711028090110508, 13.02338356790365178018126372172, 13.70502625107856088364088230130, 15.59295542156338673939578179420, 16.61026595290830721937027171266, 17.44775377427845026446345512801, 18.27999948433194175918558108689, 19.22729334767563726451817800445, 20.3539521052020306210333396182, 20.61550623785930531763889561048, 22.27435093467006023378011877009, 23.59793809816640411096894653968, 24.47972269879286153652170319527, 25.33275030212830020826599769174, 26.04813607101260456766038604343, 27.02671245302098964995191895914

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