L-function 1-105-105.74-r1-0-0

• Label: 1-105-105.74-r1-0-0
• Degree: $1$
• Conductor: $105$
• Sign: $-0.895 - 0.444i$
• Analytic cond.: $11.2838$
• Root an. cond.: $11.2838$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.5 + 0.866i)2^-s + (−0.5 − 0.866i)4^-s + 8^-s + (0.5 + 0.866i)11^-s − 13^-s + (−0.5 + 0.866i)16^-s + (−0.5 − 0.866i)17^-s + (−0.5 + 0.866i)19^-s − 22^-s + (−0.5 + 0.866i)23^-s + (0.5 − 0.866i)26^-s − 29^-s + (−0.5 − 0.866i)31^-s + (−0.5 − 0.866i)32^-s + 34^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign:	$-0.895 - 0.444i$
Analytic conductor:	\(11.2838\)
Root analytic conductor:	\(11.2838\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{105} (74, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 105,\ (1:\ ),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(-0.06533879300 + 0.2789141793i\)
\(L(1)\)	\(\approx\)	\(0.5493771009 + 0.2723412023i\)

Euler product

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

	$p$	$F_p(T)$
bad	3	\( 1 \)
	5	\( 1 \)
	7	\( 1 \)
good	2	\( 1 + (-0.5 + 0.866i)T \)
	11	\( 1 + (0.5 + 0.866i)T \)
	13	\( 1 - T \)
	17	\( 1 + (-0.5 - 0.866i)T \)
	19	\( 1 + (-0.5 + 0.866i)T \)
	23	\( 1 + (-0.5 + 0.866i)T \)
	29	\( 1 - T \)
	31	\( 1 + (-0.5 - 0.866i)T \)
	37	\( 1 + (0.5 - 0.866i)T \)

Imaginary part of the first few zeros on the critical line

−28.87548302251100609168071667510, −27.93179778206207243352133655515, −26.896822382989606730414675259218, −26.17724397322768005436317674164, −24.84344069285568189236667077100, −23.686225419326703265294688102343, −22.04601782809088400972718170663, −21.79244521856759348986315582559, −20.27544545598883976714070210571, −19.51099632786497457570814517202, −18.53881233997781784872207685699, −17.308381471025647728030947822775, −16.55318097462350856377297785837, −14.89930132395357707934589529509, −13.56924233613953713961145621733, −12.50005525862979035440925920404, −11.38762608512067854001826790455, −10.38503326779080928226001006594, −9.135631861667750098523053841659, −8.16382531935513175056389747299, −6.64928949692937849352736745075, −4.755073300678322130690741390826, −3.38397176177659754836981306601, −1.93424418653199510222221150194, −0.14096964910544635228477804826, 1.88921955558837161199913812121, 4.228694793321656657799976381, 5.49838697343202041384613665660, 6.887456644001115842130507593974, 7.78808413960726810088376298326, 9.264072879993976079549714505794, 10.02416795740906573723465796983, 11.5945825988443723407450962696, 13.05527651198810013912438132846, 14.4088456420389496956251977844, 15.14757933211887186189560277597, 16.41407208990373057305120863751, 17.32750765027055255176422823665, 18.276418771017353473044460801604, 19.449403963525748085958778709056, 20.38409126647060083294580318779, 22.0945345462781017969679419520, 22.93090849834523702737738739310, 24.084120872907276272492329698855, 25.00452398899386431373514057034, 25.82261535683761852748668378458, 27.0192396528579788513391888038, 27.687660240021414207853854844120, 28.829048007714046958839570566335, 29.89868223375412927209055191513

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