Properties

Label 2-1110-1.1-c1-0-1
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 $0$

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 − 5·11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 6·19-s − 20-s + 21-s + 5·22-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 9·29-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 − 1.50·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.67·29-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: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.357911511\)
\(L(\frac12)\) \(\approx\) \(1.357911511\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−9.834984120388578443132202063584, −8.920412853598702280342036743430, −7.996250495262613210513502734457, −7.82941954654821095457274233859, −6.78836585403012800490438370459, −5.59293502558607875381724906508, −4.63614282079891585361220034930, −3.30932495254934594571503231214, −2.50040087393469294415480604150, −0.990622762009589145457824222733, 0.990622762009589145457824222733, 2.50040087393469294415480604150, 3.30932495254934594571503231214, 4.63614282079891585361220034930, 5.59293502558607875381724906508, 6.78836585403012800490438370459, 7.82941954654821095457274233859, 7.996250495262613210513502734457, 8.920412853598702280342036743430, 9.834984120388578443132202063584

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