Properties

Label 2-1110-185.84-c1-0-3
Degree $2$
Conductor $1110$
Sign $-0.410 - 0.912i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.589 − 2.15i)5-s + 0.999·6-s + (−3.59 + 2.07i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.568 + 2.16i)10-s + 3.62·11-s + (−0.866 − 0.499i)12-s + (1.85 − 1.06i)13-s + 4.15·14-s + (1.58 + 1.57i)15-s + (−0.5 + 0.866i)16-s + (1.00 + 0.577i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.263 − 0.964i)5-s + 0.408·6-s + (−1.36 + 0.785i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.179 + 0.683i)10-s + 1.09·11-s + (−0.249 − 0.144i)12-s + (0.513 − 0.296i)13-s + 1.11·14-s + (0.410 + 0.406i)15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.410 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.410 - 0.912i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.410 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2432164707\)
\(L(\frac12)\) \(\approx\) \(0.2432164707\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.589 + 2.15i)T \)
37 \( 1 + (6.08 - 0.0539i)T \)
good7 \( 1 + (3.59 - 2.07i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.62T + 11T^{2} \)
13 \( 1 + (-1.85 + 1.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.00 - 0.577i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.73 + 6.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.363iT - 23T^{2} \)
29 \( 1 + 5.15T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
41 \( 1 + (-6.04 - 10.4i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 - 2.21iT - 47T^{2} \)
53 \( 1 + (9.47 + 5.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.12 + 5.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.3 - 7.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0732 + 0.126i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.63iT - 73T^{2} \)
79 \( 1 + (-7.85 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.9 - 6.91i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.67 - 9.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.818277724901191877393381719696, −9.267254062457582553140124349888, −8.863246847885473061162629608125, −7.83675965842080979367836180076, −6.52021645266334240559722275153, −6.13210267315407874977701589029, −4.86448072382082013532672399926, −3.88108000731615232354679712623, −2.87020064969241422549800957388, −1.23588312067007162708139523004, 0.15463477440082842031933537963, 1.78411604426560894181251395393, 3.43177465491777401894082320877, 4.06156664112276904839467984626, 5.89358431594469440313934392896, 6.35235662912018684985531139441, 7.02874096086844997717982868224, 7.64352749347480788750258329777, 8.832438576169907987665504716254, 9.656830222087631194942946719484

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