Properties

Label 2-1110-1.1-c3-0-52
Degree $2$
Conductor $1110$
Sign $-1$
Analytic cond. $65.4921$
Root an. cond. $8.09272$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s − 5·5-s − 6·6-s + 7-s − 8·8-s + 9·9-s + 10·10-s − 47·11-s + 12·12-s + 35·13-s − 2·14-s − 15·15-s + 16·16-s + 55·17-s − 18·18-s − 125·19-s − 20·20-s + 3·21-s + 94·22-s + 213·23-s − 24·24-s + 25·25-s − 70·26-s + 27·27-s + 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.0539·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.28·11-s + 0.288·12-s + 0.746·13-s − 0.0381·14-s − 0.258·15-s + 1/4·16-s + 0.784·17-s − 0.235·18-s − 1.50·19-s − 0.223·20-s + 0.0311·21-s + 0.910·22-s + 1.93·23-s − 0.204·24-s + 1/5·25-s − 0.528·26-s + 0.192·27-s + 0.0269·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(65.4921\)
Root analytic conductor: \(8.09272\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1110,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 - p T \)
5 \( 1 + p T \)
37 \( 1 + p T \)
good7 \( 1 - T + p^{3} T^{2} \)
11 \( 1 + 47 T + p^{3} T^{2} \)
13 \( 1 - 35 T + p^{3} T^{2} \)
17 \( 1 - 55 T + p^{3} T^{2} \)
19 \( 1 + 125 T + p^{3} T^{2} \)
23 \( 1 - 213 T + p^{3} T^{2} \)
29 \( 1 - 6 T + p^{3} T^{2} \)
31 \( 1 + 98 T + p^{3} T^{2} \)
41 \( 1 + 40 T + p^{3} T^{2} \)
43 \( 1 + 188 T + p^{3} T^{2} \)
47 \( 1 - 304 T + p^{3} T^{2} \)
53 \( 1 - 341 T + p^{3} T^{2} \)
59 \( 1 + 518 T + p^{3} T^{2} \)
61 \( 1 + 382 T + p^{3} T^{2} \)
67 \( 1 + 578 T + p^{3} T^{2} \)
71 \( 1 - 882 T + p^{3} T^{2} \)
73 \( 1 + 713 T + p^{3} T^{2} \)
79 \( 1 - 288 T + p^{3} T^{2} \)
83 \( 1 + 849 T + p^{3} T^{2} \)
89 \( 1 + 1263 T + p^{3} T^{2} \)
97 \( 1 + 1756 T + p^{3} T^{2} \)
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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

−8.820106579355200427029914785612, −8.372546136208316619855041792795, −7.57940303953154126986597038216, −6.85724978682850641504227787602, −5.71912807900745784806551609284, −4.65254839303622597645582772932, −3.43255804732070327599116390933, −2.61318612093453186156253996926, −1.35105572888509436650987716739, 0, 1.35105572888509436650987716739, 2.61318612093453186156253996926, 3.43255804732070327599116390933, 4.65254839303622597645582772932, 5.71912807900745784806551609284, 6.85724978682850641504227787602, 7.57940303953154126986597038216, 8.372546136208316619855041792795, 8.820106579355200427029914785612

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