L-function 1-960-960.149-r1-0-0

• Label: 1-960-960.149-r1-0-0
• Degree: $1$
• Conductor: $960$
• Sign: $-0.0980 + 0.995i$
• Analytic cond.: $103.166$
• Root an. cond.: $103.166$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.707 − 0.707i)7^-s + (−0.923 − 0.382i)11^-s + (−0.382 − 0.923i)13^-s − i·17^-s + (−0.382 − 0.923i)19^-s + (−0.707 + 0.707i)23^-s + (−0.923 + 0.382i)29^-s − 31^-s + (0.382 − 0.923i)37^-s + (0.707 − 0.707i)41^-s + (0.923 + 0.382i)43^-s − i·47^-s + i·49^-s + (0.923 + 0.382i)53^-s + (−0.382 + 0.923i)59^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$960$    =    $2^{6} \cdot 3 \cdot 5$
Sign:	$-0.0980 + 0.995i$
Analytic conductor:	$103.166$
Root analytic conductor:	$103.166$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{960} (149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 960,\ (1:\ ),\ -0.0980 + 0.995i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.02590577650 + 0.02858261975i$
$L(1)$	$\approx$	$0.7166411420 - 0.1971630242i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1$
good	7	$1 + (-0.707 - 0.707i)T$
	11	$1 + (-0.923 - 0.382i)T$
	13	$1 + (-0.382 - 0.923i)T$
	17	$1 - iT$
	19	$1 + (-0.382 - 0.923i)T$
	23	$1 + (-0.707 + 0.707i)T$
	29	$1 + (-0.923 + 0.382i)T$
	31	$1 - T$
	37	$1 + (0.382 - 0.923i)T$

Imaginary part of the first few zeros on the critical line

−21.43171932609442704427936477920, −20.63684401644699843741223288739, −19.74123287844023692248277045787, −18.79700531270012329770135933286, −18.5373718156732452553684221944, −17.375887278074413008193528473083, −16.52543251236769277142581625780, −15.905577086193338636528337370464, −14.95816803432272622298912775780, −14.38703231796498427744080334038, −13.12871264110494473233607005787, −12.63204354750149239897851342689, −11.87055034382066077966037110008, −10.78380250079415858403795175053, −9.97174674345307903595237579153, −9.24053497767353320896168914333, −8.27860137865205904515190315776, −7.45924023919548246715642995877, −6.31542054543623052941207211602, −5.76670312748212551589792240881, −4.587226936832344518521841953494, −3.6902900695761128297500497267, −2.49928280401436692489195515697, −1.79312453441752566872084345835, −0.01114976030804635271047849263, 0.69963458257967068332028789585, 2.32798489827177614708864504692, 3.149823549660015910562385769303, 4.10591088818822680197788264416, 5.26421827544306636932709589560, 5.93378046354573909022881788244, 7.35141821987915836058023954841, 7.48259663386635322167895698208, 8.871204576014224662477527331195, 9.643015390740933743274532515776, 10.55060207750226950020376581001, 11.10695591642189133368712008613, 12.316285365921652730122152081, 13.12563507860693469698806374736, 13.60141556986756421115378192141, 14.6349488083046924239256629084, 15.63205492976793005672192696766, 16.14954114202282741346647559013, 17.02393286897003972712683723867, 17.915523771167833420624494548206, 18.54253639865537868062910119453, 19.69368639532925152614443634629, 20.01404315138393070833172817255, 20.9625702773102534000983152954, 21.82868496028036918791643513969

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