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Properties

Label 2-1110-111.80-c1-0-17
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: $\chi_{1110} (191, \cdot )$
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

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

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