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Properties

Label 2-1110-185.117-c1-0-22
Degree $2$
Conductor $1110$
Sign $0.724 - 0.688i$
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 + (0.707 + 0.707i)3-s + 4-s + (2.21 − 0.325i)5-s + (0.707 + 0.707i)6-s + (0.597 + 0.597i)7-s + 8-s + 1.00i·9-s + (2.21 − 0.325i)10-s + 6.22i·11-s + (0.707 + 0.707i)12-s − 0.986·13-s + (0.597 + 0.597i)14-s + (1.79 + 1.33i)15-s + 16-s − 5.73i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (0.989 − 0.145i)5-s + (0.288 + 0.288i)6-s + (0.225 + 0.225i)7-s + 0.353·8-s + 0.333i·9-s + (0.699 − 0.102i)10-s + 1.87i·11-s + (0.204 + 0.204i)12-s − 0.273·13-s + (0.159 + 0.159i)14-s + (0.463 + 0.344i)15-s + 0.250·16-s − 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.724 - 0.688i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.724 - 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.411573693\)
\(L(\frac12)\) \(\approx\) \(3.411573693\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-2.21 + 0.325i)T \)
37 \( 1 + (0.865 + 6.02i)T \)
good7 \( 1 + (-0.597 - 0.597i)T + 7iT^{2} \)
11 \( 1 - 6.22iT - 11T^{2} \)
13 \( 1 + 0.986T + 13T^{2} \)
17 \( 1 + 5.73iT - 17T^{2} \)
19 \( 1 + (1.99 - 1.99i)T - 19iT^{2} \)
23 \( 1 + 1.91T + 23T^{2} \)
29 \( 1 + (-3.12 - 3.12i)T + 29iT^{2} \)
31 \( 1 + (0.922 - 0.922i)T - 31iT^{2} \)
41 \( 1 + 1.55iT - 41T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (2.10 + 2.10i)T + 47iT^{2} \)
53 \( 1 + (-9.01 + 9.01i)T - 53iT^{2} \)
59 \( 1 + (-5.56 + 5.56i)T - 59iT^{2} \)
61 \( 1 + (1.20 - 1.20i)T - 61iT^{2} \)
67 \( 1 + (-9.76 + 9.76i)T - 67iT^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + (-8.62 - 8.62i)T + 73iT^{2} \)
79 \( 1 + (-4.54 + 4.54i)T - 79iT^{2} \)
83 \( 1 + (7.29 - 7.29i)T - 83iT^{2} \)
89 \( 1 + (3.84 + 3.84i)T + 89iT^{2} \)
97 \( 1 - 12.5iT - 97T^{2} \)
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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

−9.917353218684666865878888356198, −9.361128970627246604759897327996, −8.347428337109570018933647885340, −7.22027526381639264671856952707, −6.63903339486135804596949390375, −5.23141140570184402567723239873, −4.98784840930775772178756323040, −3.87383201558994387388863034952, −2.50348675961931663073200463628, −1.88952434646541915973343017293, 1.28099689221688738182945908242, 2.48834323703715481514051655591, 3.36460789569390828339248926094, 4.47396807214681214152823286610, 5.78284500463007863993585911841, 6.11796966978054158164334928416, 7.03691286124222483903353692568, 8.253352937431956823367099006647, 8.656428207004610993815222498144, 9.877717544114998789048168311372

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