L-function 1-104-104.5-r1-0-0

• Label: 1-104-104.5-r1-0-0
• Degree: $1$
• Conductor: $104$
• Sign: $-0.957 - 0.289i$
• Analytic cond.: $11.1763$
• Root an. cond.: $11.1763$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 3^-s + i·5^-s + i·7^-s + 9^-s − i·11^-s − i·15^-s − 17^-s + i·19^-s − i·21^-s − 23^-s − 25^-s − 27^-s − 29^-s − i·31^-s + i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(104\)    =    \(2^{3} \cdot 13\)
Sign:	$-0.957 - 0.289i$
Analytic conductor:	\(11.1763\)
Root analytic conductor:	\(11.1763\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{104} (5, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 104,\ (1:\ ),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(-0.03549308692 + 0.2397068222i\)
\(L(1)\)	\(\approx\)	\(0.5896806770 + 0.1785409430i\)

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−28.74773727419156759245152690341, −28.12671862199252356156213723112, −27.12126330723912518095216256273, −25.92039501628253229625300199782, −24.446055307379791977571898482055, −23.78987785580122184807518046961, −22.84887163546698866287452069959, −21.758947770471268121026484517984, −20.48313951514751819791611690192, −19.75698175345698899749115838607, −17.97012514052447120862961398462, −17.299105567099020438728504314346, −16.37886346450820465072081248230, −15.36988401689207659325177850398, −13.55101827183204449210462405576, −12.711375058735052779432976724393, −11.59911162948404428191720909436, −10.41960509504470338642998829976, −9.30350124558988017840851960786, −7.628570257591711888304802724853, −6.50797872571950845249149955906, −4.946625639809738240268094188345, −4.224102755157359037302939845538, −1.58277320383327534310472232191, −0.118225366233975235185216207309, 2.16408054942311863700230737045, 3.86041191888646508791778259121, 5.68198137517492393137444780698, 6.293479942169842924520633483766, 7.78006975434942007726717894021, 9.4119398685242990447046346503, 10.770843224219191015580802842227, 11.47952369454014867724831207300, 12.62178936255578844442337313251, 14.06438290959327229607127453354, 15.35064383048127301623231057123, 16.21103371935390479088763262190, 17.57610256422204852576705240079, 18.49022554263852922285608734677, 19.13521263686051600045093390449, 21.02930947074692448592870926761, 22.16327900189866611507259194817, 22.4253448145791674100649335163, 23.865192902755416495838185301802, 24.71980946370176285671740514301, 26.105397810164068015539884279575, 27.10378292513532574349387336555, 28.03006693351008376914981285162, 29.14091306439309205881993268738, 29.818799709033310369186178057784

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