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Properties

Label 2-1110-111.68-c1-0-28
Degree $2$
Conductor $1110$
Sign $0.868 - 0.494i$
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 + (0.349 + 1.69i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−1.44 − 0.952i)6-s + 2.97·7-s + (0.707 + 0.707i)8-s + (−2.75 + 1.18i)9-s + 1.00·10-s + 2.43·11-s + (1.69 − 0.349i)12-s + (3.37 − 3.37i)13-s + (−2.10 + 2.10i)14-s + (0.952 − 1.44i)15-s − 1.00·16-s + (−2.64 − 2.64i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.201 + 0.979i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (−0.590 − 0.388i)6-s + 1.12·7-s + (0.250 + 0.250i)8-s + (−0.918 + 0.395i)9-s + 0.316·10-s + 0.735·11-s + (0.489 − 0.100i)12-s + (0.937 − 0.937i)13-s + (−0.562 + 0.562i)14-s + (0.245 − 0.373i)15-s − 0.250·16-s + (−0.640 − 0.640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.868 - 0.494i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.868 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.465220507\)
\(L(\frac12)\) \(\approx\) \(1.465220507\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.349 - 1.69i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-5.17 - 3.19i)T \)
good7 \( 1 - 2.97T + 7T^{2} \)
11 \( 1 - 2.43T + 11T^{2} \)
13 \( 1 + (-3.37 + 3.37i)T - 13iT^{2} \)
17 \( 1 + (2.64 + 2.64i)T + 17iT^{2} \)
19 \( 1 + (-5.18 + 5.18i)T - 19iT^{2} \)
23 \( 1 + (4.56 + 4.56i)T + 23iT^{2} \)
29 \( 1 + (-2.17 + 2.17i)T - 29iT^{2} \)
31 \( 1 + (1.20 + 1.20i)T + 31iT^{2} \)
41 \( 1 - 7.04T + 41T^{2} \)
43 \( 1 + (3.83 - 3.83i)T - 43iT^{2} \)
47 \( 1 + 6.56iT - 47T^{2} \)
53 \( 1 + 0.00856iT - 53T^{2} \)
59 \( 1 + (3.57 + 3.57i)T + 59iT^{2} \)
61 \( 1 + (-8.28 - 8.28i)T + 61iT^{2} \)
67 \( 1 - 3.64iT - 67T^{2} \)
71 \( 1 - 9.63iT - 71T^{2} \)
73 \( 1 + 5.75iT - 73T^{2} \)
79 \( 1 + (-6.34 + 6.34i)T - 79iT^{2} \)
83 \( 1 - 16.7iT - 83T^{2} \)
89 \( 1 + (1.72 - 1.72i)T - 89iT^{2} \)
97 \( 1 + (4.30 - 4.30i)T - 97iT^{2} \)
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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

−9.715864569485965583653843723466, −9.013537983098865598587042313621, −8.310121063102071708941737032955, −7.79934635509937543801552275659, −6.57304532979668261018764570903, −5.50843389382231385988648060030, −4.76793226851745737528055834778, −3.98359430494789411866170584024, −2.60515598025642607573449150635, −0.883078748690873460733851363665, 1.33597222424802001039519866226, 1.91349824106813680691804316580, 3.41562935004372967778499785183, 4.21539845998096687133345309424, 5.75304017939861546909042650533, 6.59117290155180514680693910737, 7.57427305322679544959099993400, 8.082398706037111102982792851960, 8.843210875838801915007425733514, 9.621111583518854360411604188387

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