Properties

Label 2-1110-111.68-c1-0-11
Degree $2$
Conductor $1110$
Sign $0.219 - 0.975i$
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  + (−0.707 + 0.707i)2-s + (−1.59 − 0.670i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (1.60 − 0.655i)6-s + 3.28·7-s + (0.707 + 0.707i)8-s + (2.10 + 2.14i)9-s + 1.00·10-s − 1.74·11-s + (−0.670 + 1.59i)12-s + (−3.25 + 3.25i)13-s + (−2.32 + 2.32i)14-s + (0.655 + 1.60i)15-s − 1.00·16-s + (−1.24 − 1.24i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.922 − 0.387i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (0.654 − 0.267i)6-s + 1.24·7-s + (0.250 + 0.250i)8-s + (0.700 + 0.713i)9-s + 0.316·10-s − 0.526·11-s + (−0.193 + 0.461i)12-s + (−0.903 + 0.903i)13-s + (−0.621 + 0.621i)14-s + (0.169 + 0.414i)15-s − 0.250·16-s + (−0.301 − 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7164795298\)
\(L(\frac12)\) \(\approx\) \(0.7164795298\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.59 + 0.670i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (4.68 + 3.87i)T \)
good7 \( 1 - 3.28T + 7T^{2} \)
11 \( 1 + 1.74T + 11T^{2} \)
13 \( 1 + (3.25 - 3.25i)T - 13iT^{2} \)
17 \( 1 + (1.24 + 1.24i)T + 17iT^{2} \)
19 \( 1 + (-0.258 + 0.258i)T - 19iT^{2} \)
23 \( 1 + (-1.64 - 1.64i)T + 23iT^{2} \)
29 \( 1 + (-0.378 + 0.378i)T - 29iT^{2} \)
31 \( 1 + (-3.64 - 3.64i)T + 31iT^{2} \)
41 \( 1 - 9.92T + 41T^{2} \)
43 \( 1 + (4.35 - 4.35i)T - 43iT^{2} \)
47 \( 1 - 2.28iT - 47T^{2} \)
53 \( 1 - 9.35iT - 53T^{2} \)
59 \( 1 + (0.666 + 0.666i)T + 59iT^{2} \)
61 \( 1 + (0.134 + 0.134i)T + 61iT^{2} \)
67 \( 1 - 8.70iT - 67T^{2} \)
71 \( 1 - 7.14iT - 71T^{2} \)
73 \( 1 - 13.7iT - 73T^{2} \)
79 \( 1 + (-6.38 + 6.38i)T - 79iT^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + (-0.495 + 0.495i)T - 89iT^{2} \)
97 \( 1 + (-4.16 + 4.16i)T - 97iT^{2} \)
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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

−10.03778317616772920572757312835, −9.096259433630768725740848990713, −8.181079552811431207181244678956, −7.47167614009085847607687709486, −6.91227345362679515508163308193, −5.75970153570031359523108328586, −4.92309010426849651702358231988, −4.44518283736301698711752698328, −2.27435314520566136975861978455, −1.11438919702455849439798258988, 0.49905499513304203548585446977, 2.04724470064602700602446805783, 3.35601105327193413611975495988, 4.60616143313006017503623493431, 5.11314025556821681126467805491, 6.28982317931440694054574054082, 7.41658772246871942363348919599, 7.960170905336503858068262236219, 8.915716989953105673685970017223, 10.04420395881247012276959717375

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