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Properties

Label 2-1110-185.184-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.280 + 0.959i$
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 + (−2.23 + 0.0950i)5-s − i·6-s + 3.20i·7-s − 8-s − 9-s + (2.23 − 0.0950i)10-s − 3.73·11-s + i·12-s + 0.424·13-s − 3.20i·14-s + (−0.0950 − 2.23i)15-s + 16-s − 6.90·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.999 + 0.0425i)5-s − 0.408i·6-s + 1.21i·7-s − 0.353·8-s − 0.333·9-s + (0.706 − 0.0300i)10-s − 1.12·11-s + 0.288i·12-s + 0.117·13-s − 0.857i·14-s + (−0.0245 − 0.576i)15-s + 0.250·16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.280 + 0.959i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.280 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3020936297\)
\(L(\frac12)\) \(\approx\) \(0.3020936297\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0950i)T \)
37 \( 1 + (-5.90 + 1.45i)T \)
good7 \( 1 - 3.20iT - 7T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 - 0.424T + 13T^{2} \)
17 \( 1 + 6.90T + 17T^{2} \)
19 \( 1 + 8.02iT - 19T^{2} \)
23 \( 1 - 6.34T + 23T^{2} \)
29 \( 1 - 3.39iT - 29T^{2} \)
31 \( 1 - 1.20iT - 31T^{2} \)
41 \( 1 - 5.74T + 41T^{2} \)
43 \( 1 + 9.31T + 43T^{2} \)
47 \( 1 - 2.04iT - 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 + 2.16iT - 59T^{2} \)
61 \( 1 + 1.45iT - 61T^{2} \)
67 \( 1 + 11.4iT - 67T^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 - 4.50iT - 79T^{2} \)
83 \( 1 + 7.00iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 - 4.43T + 97T^{2} \)
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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

−9.414649604601669438310470580675, −8.823870762994090661242419414014, −8.362367273170359784737396709188, −7.27139926452396677185915392440, −6.52077852440277858066546832098, −5.20108730770151976255151072944, −4.61654395462939448843453810250, −3.09225472628970756869268564914, −2.43346613747962732211237618972, −0.19431395075347791987408952303, 1.07368554555471315914285004436, 2.56937586563418817682063556446, 3.76833216999829774679003071241, 4.71096534885493830374000040041, 6.08867155112195702470790604333, 7.02249330634746044048974172356, 7.63778061793114876802823376007, 8.146201157612888196975945891093, 9.003639554745667505053729124283, 10.18985496673773360101505345876

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