↑

Properties

Label 2-1110-185.174-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.958 + 0.285i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (−2.23 − 0.133i)5-s + 0.999·6-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.99 + i)10-s + (0.866 − 0.499i)12-s + (4.33 + 2.5i)13-s + 0.999·14-s + (−1.86 − 1.23i)15-s + (−0.5 − 0.866i)16-s + (1.73 − i)17-s + (0.866 + 0.499i)18-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.998 − 0.0599i)5-s + 0.408·6-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.632 + 0.316i)10-s + (0.249 − 0.144i)12-s + (1.20 + 0.693i)13-s + 0.267·14-s + (−0.481 − 0.318i)15-s + (−0.125 − 0.216i)16-s + (0.420 − 0.242i)17-s + (0.204 + 0.117i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.958 + 0.285i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.958 + 0.285i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.594842009\)
\(L(\frac12)\) \(\approx\) \(2.594842009\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (2.23 + 0.133i)T \)
37 \( 1 + (-6.06 - 0.5i)T \)
good7 \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-4.33 - 2.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + (6.92 - 4i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.5 + 11.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 - 4.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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

−9.744456368489773586054221668569, −9.018701526505663771229411549470, −8.195324366155719898409198998959, −7.36581966562438393814904948628, −6.43794342097288530993385996286, −5.20707804979833857501634806599, −4.44222163987766319910492261893, −3.59036119995269330384827213523, −2.76816175974931117016177013765, −1.21645208584859894192507603955, 1.21616380056785617395438266959, 2.94286844418699668156778010992, 3.68462232880099444160187359942, 4.49767442382488319068266508469, 5.69745023545524531960227910951, 6.52618045543750663076498260326, 7.57321177287014765497699792821, 8.092528271600299982367993819172, 8.595571718308044492730378260959, 9.944083564098184151010413792838

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