↑

Properties

Label 2-110-11.7-c2-0-2
Degree $2$
Conductor $110$
Sign $0.702 - 0.711i$
Analytic cond. $2.99728$
Root an. cond. $1.73126$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.831 + 1.14i)2-s + (−0.138 + 0.426i)3-s + (−0.618 − 1.90i)4-s + (1.80 − 1.31i)5-s + (−0.372 − 0.513i)6-s + (6.37 − 2.07i)7-s + (2.68 + 0.874i)8-s + (7.11 + 5.17i)9-s + 3.16i·10-s + (−0.896 + 10.9i)11-s + 0.897·12-s + (0.146 − 0.201i)13-s + (−2.93 + 9.02i)14-s + (0.309 + 0.954i)15-s + (−3.23 + 2.35i)16-s + (−6.00 − 8.26i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.0462 + 0.142i)3-s + (−0.154 − 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.0621 − 0.0855i)6-s + (0.911 − 0.296i)7-s + (0.336 + 0.109i)8-s + (0.790 + 0.574i)9-s + 0.316i·10-s + (−0.0814 + 0.996i)11-s + 0.0747·12-s + (0.0112 − 0.0154i)13-s + (−0.209 + 0.644i)14-s + (0.0206 + 0.0636i)15-s + (−0.202 + 0.146i)16-s + (−0.353 − 0.486i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.702 - 0.711i$
Analytic conductor: \(2.99728\)
Root analytic conductor: \(1.73126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1),\ 0.702 - 0.711i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17614 + 0.491292i\)
\(L(\frac12)\) \(\approx\) \(1.17614 + 0.491292i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.831 - 1.14i)T \)
5 \( 1 + (-1.80 + 1.31i)T \)
11 \( 1 + (0.896 - 10.9i)T \)
good3 \( 1 + (0.138 - 0.426i)T + (-7.28 - 5.29i)T^{2} \)
7 \( 1 + (-6.37 + 2.07i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (-0.146 + 0.201i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (6.00 + 8.26i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-31.7 - 10.3i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 5.46T + 529T^{2} \)
29 \( 1 + (-17.4 + 5.67i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (29.3 + 21.3i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (19.0 + 58.5i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (14.1 + 4.60i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 26.3iT - 1.84e3T^{2} \)
47 \( 1 + (26.0 - 80.1i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (71.6 + 52.0i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-17.8 - 54.8i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (9.92 + 13.6i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 59.7T + 4.48e3T^{2} \)
71 \( 1 + (101. - 73.8i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (23.9 - 7.78i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-47.0 + 64.8i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (6.26 + 8.61i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 41.9T + 7.92e3T^{2} \)
97 \( 1 + (130. + 94.8i)T + (2.90e3 + 8.94e3i)T^{2} \)
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

−13.79034579131788461066124032290, −12.62094108708202122706502858683, −11.28155154011972542853680076719, −10.14140739311015275119829142983, −9.351916203218076319946061826301, −7.86582068012743243366523392495, −7.18421000828767092895254319554, −5.43566417722656709149077471751, −4.46866682808920166592520279364, −1.68654443943407795532145367362, 1.44039662789522830895469081336, 3.30146643485026394677773941786, 5.07471896378010991062141998691, 6.66219796090299878732863816739, 7.994345094509799715827179626365, 9.070447672442392642453840995322, 10.16665994886369352255149697573, 11.23626171853902673791797071483, 12.01634746610776046446090907590, 13.24512864586868814295195398430

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