L-function 1-1037-1037.150-r0-0-0

• Label: 1-1037-1037.150-r0-0-0
• Degree: $1$
• Conductor: $1037$
• Sign: $-0.932 - 0.360i$
• Analytic cond.: $4.81580$
• Root an. cond.: $4.81580$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.156 − 0.987i)2^-s + (−0.522 − 0.852i)3^-s + (−0.951 + 0.309i)4^-s + (−0.996 − 0.0784i)5^-s + (−0.760 + 0.649i)6^-s + (0.522 − 0.852i)7^-s + (0.453 + 0.891i)8^-s + (−0.453 + 0.891i)9^-s + (0.0784 + 0.996i)10^-s + (−0.382 − 0.923i)11^-s + (0.760 + 0.649i)12^-s + i·13^-s + (−0.923 − 0.382i)14^-s + (0.453 + 0.891i)15^-s + (0.809 − 0.587i)16^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1037$    =    $17 \cdot 61$
Sign:	$-0.932 - 0.360i$
Analytic conductor:	$4.81580$
Root analytic conductor:	$4.81580$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1037} (150, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1037,\ (0:\ ),\ -0.932 - 0.360i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1530677705 - 0.8214973698i$
$L(1)$	$\approx$	$0.4719931686 - 0.5284170073i$

Euler product

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

	$p$	$F_p(T)$
bad	17	$1$
	61	$1$
good	2	$1 + (-0.156 - 0.987i)T$
	3	$1 + (-0.522 - 0.852i)T$
	5	$1 + (-0.996 - 0.0784i)T$
	7	$1 + (0.522 - 0.852i)T$
	11	$1 + (-0.382 - 0.923i)T$
	13	$1 + iT$
	19	$1 + (0.987 - 0.156i)T$
	23	$1 + (0.760 - 0.649i)T$
	29	$1 + (0.923 + 0.382i)T$

Imaginary part of the first few zeros on the critical line

−22.26862764929330378522994801358, −21.23196462664769700471638639657, −20.4148516355519187697774503219, −19.50453357780224514213607988826, −18.42098754919725348508838563474, −17.89849182653043962386773257384, −17.18291635260628515547974463958, −16.20372984641137239892220889746, −15.467482124736460751598172727564, −15.22635419015560316681329791579, −14.58357866643902886337039618045, −13.2632654970011922881217668178, −12.226274788908676939656812183194, −11.63154903273800464022519764819, −10.55514921023072439313532432501, −9.800579185236006783974289775748, −8.92366563503038550577394829920, −8.05136930721501348514628871477, −7.42296195906537877197795213573, −6.292757307728278302584333640094, −5.261366454407269770916939435048, −4.91059390600793415014070354069, −3.912389189844740468765161962947, −2.87456883693361406846436245251, −0.89272911749898654170547959496, 0.63769684759425336005578469391, 1.28671823986827524200920432831, 2.61203714421052105948077633826, 3.5762253775004045069791996864, 4.58931868001169445532923954282, 5.26055981090609087387706169013, 6.750403935420778139571691312075, 7.50365507097528476887159832147, 8.278319274673209124913015423459, 8.97365114263555143984159962612, 10.47559621693869222793042614243, 10.98228006335082357560835144888, 11.636650583664812562160655949073, 12.23480996880150815400170356324, 13.16323390647354483220108650402, 13.905998197881607405291724609462, 14.48396789889850662161239127138, 16.21392694929473899660294740840, 16.52371641216729207777104155681, 17.622997028210593885844330426887, 18.19477599467491574310155991553, 19.176372412619710873652283721064, 19.38710416883169164123468397795, 20.321642817802683666820980307197, 21.09963635014182478176790177171

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