Properties

Label 2-111090-1.1-c1-0-57
Degree $2$
Conductor $111090$
Sign $-1$
Analytic cond. $887.058$
Root an. cond. $29.7835$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s − 14-s + 15-s + 16-s − 6·17-s + 18-s + 8·19-s − 20-s + 21-s + 6·22-s − 24-s + 25-s − 27-s − 28-s − 6·29-s + 30-s − 6·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.27·22-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111090\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(887.058\)
Root analytic conductor: \(29.7835\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{111090} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 111090,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.97387375650940, −13.42921069791260, −12.87767507218053, −12.26279211477349, −12.02826315750208, −11.55056613675241, −11.10844755303044, −10.77346288984646, −9.980968992962969, −9.470399184599770, −8.938390742425730, −8.690246932335736, −7.550398292813925, −7.237288347448858, −6.930209020958495, −6.185175123295208, −5.946216505387127, −5.114841665356467, −4.807593901220502, −3.911587974125185, −3.755844836536223, −3.219215862310078, −2.232567920609283, −1.616260207443861, −0.9294441300077686, 0, 0.9294441300077686, 1.616260207443861, 2.232567920609283, 3.219215862310078, 3.755844836536223, 3.911587974125185, 4.807593901220502, 5.114841665356467, 5.946216505387127, 6.185175123295208, 6.930209020958495, 7.237288347448858, 7.550398292813925, 8.690246932335736, 8.938390742425730, 9.470399184599770, 9.980968992962969, 10.77346288984646, 11.10844755303044, 11.55056613675241, 12.02826315750208, 12.26279211477349, 12.87767507218053, 13.42921069791260, 13.97387375650940

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