L-function 1-111-111.11-r1-0-0

• Label: 1-111-111.11-r1-0-0
• Degree: $1$
• Conductor: $111$
• Sign: $-0.729 + 0.683i$
• Analytic cond.: $11.9286$
• Root an. cond.: $11.9286$
• 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 + (−0.5 − 0.866i)5^-s + (−0.5 − 0.866i)7^-s + 8^-s + 10^-s − 11^-s + (0.5 + 0.866i)13^-s + 14^-s + (−0.5 + 0.866i)16^-s + (−0.5 + 0.866i)17^-s + (0.5 + 0.866i)19^-s + (−0.5 + 0.866i)20^-s + (0.5 − 0.866i)22^-s + 23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(111\)    =    \(3 \cdot 37\)
Sign:	$-0.729 + 0.683i$
Analytic conductor:	\(11.9286\)
Root analytic conductor:	\(11.9286\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{111} (11, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 111,\ (1:\ ),\ -0.729 + 0.683i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.1731861736 + 0.4380510773i\)
\(L(1)\)	\(\approx\)	\(0.5707492847 + 0.1729354615i\)

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−28.84428663386952894681521142245, −27.85775951776313323296385540413, −26.874721840709850194989140379752, −26.01646822791616561491244688949, −25.08246981753121429901743269624, −23.280948149783554203045088007977, −22.46680978889936320127330618634, −21.59357804537094595569203226822, −20.38063017089029831450835560418, −19.37195830253484194471561030649, −18.38440996540105037020399430683, −17.895670205132920454622593730445, −16.06680036638066230645093425222, −15.301835065179126544002326318415, −13.59645496759416804588355412209, −12.59329291259381387452061187711, −11.39084289453306125537361127240, −10.58417595354474054757725152949, −9.33740184349817230579012444655, −8.14723827609410907726705191324, −6.94349535581063222244872477636, −5.1128225659239376604928878020, −3.25584682356959233363991786569, −2.57454902444139282785457736245, −0.26300186164647879883209150835, 1.25586006238920775137270620587, 3.91373795285137470312594635368, 5.09753157102330768031997573950, 6.535002557450403002290483829517, 7.723358531750207021076452668818, 8.6774020710314261146334382238, 9.88896605572155668735347697419, 11.05381150842799778656207440015, 12.83448656502186152301905084661, 13.67194533477038955328420984284, 15.08875887312005374095422311231, 16.268627024369608843898344604916, 16.6365301154778564992366542986, 17.97282616698046090872956010317, 19.16380876021688473483402447464, 20.00750805413261850280811574514, 21.19206251001443005133338936322, 22.99271772663079980925804960130, 23.551263700959713101395058545309, 24.39869508876239726169160887144, 25.62475146203557950622617304875, 26.515712996757585624241992378643, 27.29992989447889187139220747280, 28.63003529431762194146403431096, 29.00228315503309361397754236332

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