L-function 1-1037-1037.112-r0-0-0

• Label: 1-1037-1037.112-r0-0-0
• Degree: $1$
• Conductor: $1037$
• Sign: $0.862 - 0.505i$
• 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

+ (−0.998 − 0.0523i)2^-s + (−0.996 + 0.0784i)3^-s + (0.994 + 0.104i)4^-s + (−0.688 + 0.725i)5^-s + (0.999 − 0.0261i)6^-s + (−0.566 + 0.824i)7^-s + (−0.987 − 0.156i)8^-s + (0.987 − 0.156i)9^-s + (0.725 − 0.688i)10^-s + (−0.923 − 0.382i)11^-s + (−0.999 − 0.0261i)12^-s + (0.866 + 0.5i)13^-s + (0.608 − 0.793i)14^-s + (0.629 − 0.777i)15^-s + (0.978 + 0.207i)16^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1037$    =    $17 \cdot 61$
Sign:	$0.862 - 0.505i$
Analytic conductor:	$4.81580$
Root analytic conductor:	$4.81580$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1037} (112, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1037,\ (0:\ ),\ 0.862 - 0.505i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2470004530 - 0.06706083276i$
$L(1)$	$\approx$	$0.3687522798 + 0.06607070130i$

Euler product

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

	$p$	$F_p(T)$
bad	17	$1$
	61	$1$
good	2	$1 + (-0.998 - 0.0523i)T$
	3	$1 + (-0.996 + 0.0784i)T$
	5	$1 + (-0.688 + 0.725i)T$
	7	$1 + (-0.566 + 0.824i)T$
	11	$1 + (-0.923 - 0.382i)T$
	13	$1 + (0.866 + 0.5i)T$
	19	$1 + (-0.838 - 0.544i)T$
	23	$1 + (0.522 + 0.852i)T$
	29	$1 + (-0.608 + 0.793i)T$

Imaginary part of the first few zeros on the critical line

−21.31126398807830062840412994894, −20.67545662067805817723454217876, −20.079728031789930152165915629301, −19.03289493119240834334446894165, −18.578359137825569715055967766663, −17.58261944228267770509660022255, −16.92222172080595461740778187324, −16.34734345843444976615452307452, −15.72879811980793533441660080122, −15.05457998742821483923075682595, −13.32282535851652915824590289705, −12.74499098664814059800139743951, −11.99053438451260895446166921465, −11.01015229221313570880779784484, −10.4682105371260751529323992102, −9.803069825832721454387144174490, −8.54601728350489470778162467, −7.909736898901739267830693703677, −7.02712747662318637111206637832, −6.29771425097212034400453185933, −5.28288589132837257825483922753, −4.287038193268642836880969880156, −3.20425216316921207114698850101, −1.64404163828096781815790735325, −0.68323651491672661276327847389, 0.28071766763155849603283873793, 1.82387965406517806981336923027, 2.93169913896629137975732790532, 3.827504583343122640575962979849, 5.30988379926281800205591406168, 6.17611851453107809561907280321, 6.79423006669419753578875330268, 7.65260036716748420064493860749, 8.636881232796431924143890617928, 9.48093447559694357423898818286, 10.48700979451076050249329246150, 11.07531627802898748470526211593, 11.5795924777520127063431262351, 12.4908646994228325407160102680, 13.301941923670083231287400247827, 14.96041166245723695415391291028, 15.51196196303814541738266064592, 16.09277310276087700387236838525, 16.72325634761669083272585363315, 17.91496185146638626599012474962, 18.27466396684358355356506347225, 19.11928551631994571766765633321, 19.38203623437922971543765199190, 20.8981411959712929331595364906, 21.390657142356820802991993036666

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