L-function 1-960-960.509-r1-0-0

• Label: 1-960-960.509-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:	$\chi_{960} (509, \cdot )$
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

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

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