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Properties

Label 2-1110-185.184-c1-0-16
Degree $2$
Conductor $1110$
Sign $0.415 + 0.909i$
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  − 2-s + i·3-s + 4-s + (−2.18 − 0.452i)5-s − i·6-s + 0.0776i·7-s − 8-s − 9-s + (2.18 + 0.452i)10-s + 1.27·11-s + i·12-s − 1.98·13-s − 0.0776i·14-s + (0.452 − 2.18i)15-s + 16-s + 2.25·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.979 − 0.202i)5-s − 0.408i·6-s + 0.0293i·7-s − 0.353·8-s − 0.333·9-s + (0.692 + 0.143i)10-s + 0.385·11-s + 0.288i·12-s − 0.550·13-s − 0.0207i·14-s + (0.116 − 0.565i)15-s + 0.250·16-s + 0.545·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5947480957\)
\(L(\frac12)\) \(\approx\) \(0.5947480957\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
5 \( 1 + (2.18 + 0.452i)T \)
37 \( 1 + (-4.90 + 3.59i)T \)
good7 \( 1 - 0.0776iT - 7T^{2} \)
11 \( 1 - 1.27T + 11T^{2} \)
13 \( 1 + 1.98T + 13T^{2} \)
17 \( 1 - 2.25T + 17T^{2} \)
19 \( 1 - 2.64iT - 19T^{2} \)
23 \( 1 + 7.25T + 23T^{2} \)
29 \( 1 + 5.46iT - 29T^{2} \)
31 \( 1 + 6.55iT - 31T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 - 7.27T + 43T^{2} \)
47 \( 1 + 8.23iT - 47T^{2} \)
53 \( 1 + 4.77iT - 53T^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 - 9.19iT - 61T^{2} \)
67 \( 1 - 2.98iT - 67T^{2} \)
71 \( 1 - 7.16T + 71T^{2} \)
73 \( 1 + 9.35iT - 73T^{2} \)
79 \( 1 + 15.9iT - 79T^{2} \)
83 \( 1 - 3.25iT - 83T^{2} \)
89 \( 1 - 15.4iT - 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−9.772975824275246809226197575260, −8.885836877089849137329348917528, −8.013559118106287332228722587867, −7.58894958452474974265600022676, −6.40159327042052349197662768831, −5.44713002194045266370061981901, −4.23853707421960097911344678350, −3.57361250790322123974180638119, −2.16810980539930626550784522427, −0.38149368807851886345891347969, 1.13211400822672919055381457558, 2.57136233074474605767160506507, 3.60623400671926927763453029275, 4.80376630959536572481034698149, 6.05514257170031513615720224407, 6.97128692958213124556639742304, 7.51748598368243034957117441332, 8.292423249642187652003883488038, 8.992666648488159219086250118968, 10.01225574945027493162860939840

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