↑

Properties

Label 2-1110-37.27-c1-0-26
Degree $2$
Conductor $1110$
Sign $-0.780 - 0.624i$
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 + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (0.866 − 0.5i)5-s + 0.999i·6-s + (−1.98 − 3.43i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s + 2.96·11-s + (0.499 − 0.866i)12-s + (−2.55 + 1.47i)13-s + 3.96i·14-s + (−0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + (−6.83 − 3.94i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + 0.408i·6-s + (−0.749 − 1.29i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s + 0.894·11-s + (0.144 − 0.249i)12-s + (−0.707 + 0.408i)13-s + 1.06i·14-s + (−0.223 − 0.129i)15-s + (−0.125 + 0.216i)16-s + (−1.65 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.780 - 0.624i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.780 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3443122787\)
\(L(\frac12)\) \(\approx\) \(0.3443122787\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-5.64 - 2.26i)T \)
good7 \( 1 + (1.98 + 3.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.96T + 11T^{2} \)
13 \( 1 + (2.55 - 1.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.83 + 3.94i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.384 + 0.222i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.104iT - 23T^{2} \)
29 \( 1 - 4.62iT - 29T^{2} \)
31 \( 1 + 1.74iT - 31T^{2} \)
41 \( 1 + (3.44 + 5.97i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.667iT - 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 + (5.10 - 8.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.75 + 3.32i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.97 + 3.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.63 - 2.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.77 - 6.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.09T + 73T^{2} \)
79 \( 1 + (-4.49 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.77 - 6.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (15.6 + 9.01i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.85iT - 97T^{2} \)
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

−9.452711556440034829017455588079, −8.729431673007419625317672983807, −7.52143534151960700758175637649, −6.83600350055829061382065547660, −6.43611787431535042105586381353, −4.88323035934330476056981864611, −3.98006340930006600561191174764, −2.69872396619305423535882810047, −1.45311289195724531544104690338, −0.18903805667055011395472592096, 1.96230168539105038603439968309, 3.01982661347671532958983631424, 4.39428453549898044674952000556, 5.47897338380609284034250776812, 6.32863642951085140098041553035, 6.66129930859522769160905099088, 8.105782432977871339609381757270, 8.872429212372487779504914476949, 9.543223829913197728437251728909, 9.995598997391609430224302632706

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