Properties

Label 2-1110-1.1-c1-0-24
Degree $2$
Conductor $1110$
Sign $-1$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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 − 5·7-s + 8-s + 9-s − 10-s − 5·11-s + 12-s − 13-s − 5·14-s − 15-s + 16-s − 5·17-s + 18-s − 3·19-s − 20-s − 5·21-s − 5·22-s + 3·23-s + 24-s + 25-s − 26-s + 27-s − 5·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s − 0.277·13-s − 1.33·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.688·19-s − 0.223·20-s − 1.09·21-s − 1.06·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.944·28-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: yes
Arithmetic: yes
Character: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
37 \( 1 + T \)
good7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−9.414402339797189103027728212407, −8.652925347572360710080893301069, −7.59145422677086622365283462818, −6.87125318347174773591248580776, −6.13213828431916420587586112772, −4.98150255758734417701400977273, −4.00560103951763470081398393399, −3.03535024683399848809507960758, −2.45915289451631851334035024721, 0, 2.45915289451631851334035024721, 3.03535024683399848809507960758, 4.00560103951763470081398393399, 4.98150255758734417701400977273, 6.13213828431916420587586112772, 6.87125318347174773591248580776, 7.59145422677086622365283462818, 8.652925347572360710080893301069, 9.414402339797189103027728212407

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