Properties

Label 2-1110-1.1-c1-0-6
Degree $2$
Conductor $1110$
Sign $1$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 3·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 3.81·13-s − 3·14-s + 15-s + 16-s + 17-s − 18-s + 7.81·19-s + 20-s + 3·21-s + 22-s + 5.81·23-s − 24-s + 25-s + 3.81·26-s + 27-s + 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 1.05·13-s − 0.801·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.79·19-s + 0.223·20-s + 0.654·21-s + 0.213·22-s + 1.21·23-s − 0.204·24-s + 0.200·25-s + 0.748·26-s + 0.192·27-s + 0.566·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.785379611\)
\(L(\frac12)\) \(\approx\) \(1.785379611\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
37 \( 1 - T \)
good7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 3.81T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 7.81T + 19T^{2} \)
23 \( 1 - 5.81T + 23T^{2} \)
29 \( 1 + 6.81T + 29T^{2} \)
31 \( 1 - 6.81T + 31T^{2} \)
41 \( 1 + 4.81T + 41T^{2} \)
43 \( 1 + 4.81T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 6.81T + 61T^{2} \)
67 \( 1 - 7.63T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 5.81T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 3.81T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 4.81T + 97T^{2} \)
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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

−9.773127164759084043386419234376, −9.068457563127379902859084169114, −8.173719141728138354802397577553, −7.55464997660694362544493713208, −6.87115962064833637747421136068, −5.41713309124767003739671321208, −4.86390960009462647397940138963, −3.29472424029316505048279098558, −2.30233594787457595819060582330, −1.21532669289439810366363743417, 1.21532669289439810366363743417, 2.30233594787457595819060582330, 3.29472424029316505048279098558, 4.86390960009462647397940138963, 5.41713309124767003739671321208, 6.87115962064833637747421136068, 7.55464997660694362544493713208, 8.173719141728138354802397577553, 9.068457563127379902859084169114, 9.773127164759084043386419234376

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