Properties

Label 2-111-37.11-c1-0-5
Degree $2$
Conductor $111$
Sign $0.770 + 0.637i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 − 1.14i)2-s + (−0.5 + 0.866i)3-s + (1.62 − 2.80i)4-s + (−0.483 − 0.278i)5-s + 2.28i·6-s + (−0.5 + 0.866i)7-s − 2.84i·8-s + (−0.499 − 0.866i)9-s − 1.27·10-s − 2.24·11-s + (1.62 + 2.80i)12-s + (−1.98 − 1.14i)13-s + 2.28i·14-s + (0.483 − 0.278i)15-s + (−0.0168 − 0.0292i)16-s + (−3.84 + 2.22i)17-s + ⋯
L(s)  = 1  + (1.40 − 0.809i)2-s + (−0.288 + 0.499i)3-s + (0.810 − 1.40i)4-s + (−0.216 − 0.124i)5-s + 0.934i·6-s + (−0.188 + 0.327i)7-s − 1.00i·8-s + (−0.166 − 0.288i)9-s − 0.403·10-s − 0.676·11-s + (0.468 + 0.810i)12-s + (−0.550 − 0.317i)13-s + 0.611i·14-s + (0.124 − 0.0720i)15-s + (−0.00422 − 0.00731i)16-s + (−0.933 + 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62953 - 0.586640i\)
\(L(\frac12)\) \(\approx\) \(1.62953 - 0.586640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-3.36 - 5.06i)T \)
good2 \( 1 + (-1.98 + 1.14i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.483 + 0.278i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + (1.98 + 1.14i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.84 - 2.22i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-7.33 - 4.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.02iT - 23T^{2} \)
29 \( 1 + 7.90iT - 29T^{2} \)
31 \( 1 - 1.86iT - 31T^{2} \)
41 \( 1 + (-1.87 + 3.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.28iT - 43T^{2} \)
47 \( 1 - 1.27T + 47T^{2} \)
53 \( 1 + (4.60 + 7.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.60 - 2.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.83 + 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.20 - 10.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + (-5.83 - 3.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.17 - 12.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.20 - 4.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.194iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.40613318297255700354471538835, −12.38537181779134399949738412067, −11.76531646987032119228923515161, −10.66126336543448361994264523766, −9.780014720326632796927068615128, −8.068076816160474644945851094109, −6.15944357811679605409897256288, −5.14497435042778344593379855255, −4.04366151546043772539075203826, −2.62940383735895859297154255081, 3.06716412571131950376227392492, 4.70138139200537039765081403168, 5.67181407039610043693640916696, 7.11902239030535409463362857144, 7.46541966767998478445750590813, 9.411339827819493583683354460016, 11.13465376747550016195758623224, 11.98647502192770650105367752898, 13.16006740565177467211046591371, 13.58598327103830861534653023729

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