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Properties

Label 2-1110-5.4-c1-0-8
Degree $2$
Conductor $1110$
Sign $0.987 - 0.158i$
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  − i·2-s − i·3-s − 4-s + (−0.353 − 2.20i)5-s − 6-s + 3.94i·7-s + i·8-s − 9-s + (−2.20 + 0.353i)10-s + 1.94·11-s + i·12-s + 2.85i·13-s + 3.94·14-s + (−2.20 + 0.353i)15-s + 16-s + 7.07i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.158 − 0.987i)5-s − 0.408·6-s + 1.49i·7-s + 0.353i·8-s − 0.333·9-s + (−0.698 + 0.111i)10-s + 0.587·11-s + 0.288i·12-s + 0.791i·13-s + 1.05·14-s + (−0.570 + 0.0913i)15-s + 0.250·16-s + 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.987 - 0.158i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.987 - 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.149105194\)
\(L(\frac12)\) \(\approx\) \(1.149105194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
5 \( 1 + (0.353 + 2.20i)T \)
37 \( 1 - iT \)
good7 \( 1 - 3.94iT - 7T^{2} \)
11 \( 1 - 1.94T + 11T^{2} \)
13 \( 1 - 2.85iT - 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
19 \( 1 + 1.74T + 19T^{2} \)
23 \( 1 - 4.68iT - 23T^{2} \)
29 \( 1 - 1.32T + 29T^{2} \)
31 \( 1 + 6.50T + 31T^{2} \)
41 \( 1 - 12.6T + 41T^{2} \)
43 \( 1 - 1.09iT - 43T^{2} \)
47 \( 1 - 6.70iT - 47T^{2} \)
53 \( 1 + 6.94iT - 53T^{2} \)
59 \( 1 + 2.70T + 59T^{2} \)
61 \( 1 - 3.28T + 61T^{2} \)
67 \( 1 - 1.33iT - 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + 5.08iT - 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 9.62iT - 83T^{2} \)
89 \( 1 + 2.02T + 89T^{2} \)
97 \( 1 + 1.13iT - 97T^{2} \)
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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

−9.583227406571261331007686807790, −9.015671207022849355825219738962, −8.519947236982472916111871351630, −7.63694195917959736216355303041, −6.19605039783627529005018247806, −5.69052809352075866698147595331, −4.55727870876329019867727407784, −3.60292839020152355331872349072, −2.15779988152451439268326632561, −1.46311790765117242195307524028, 0.53149171999576276133195517229, 2.79736409827427722647211706887, 3.82925535279596623241851776861, 4.49164170753902123704893951955, 5.63576228324062439523888778374, 6.68612355652844182319237979436, 7.25061305002139995710564246130, 7.908195025247750222406279862550, 9.080573042274030986790741378488, 9.836228447862045719130458700091

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