Properties

Label 2-1110-1.1-c1-0-9
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 + 3·7-s + 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 3·14-s − 15-s + 16-s − 17-s + 18-s − 5·19-s + 20-s − 3·21-s + 22-s + 7·23-s − 24-s + 25-s + 26-s − 27-s + 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s + 1.45·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-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: 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

\(L(1)\) \(\approx\) \(2.640973812\)
\(L(\frac12)\) \(\approx\) \(2.640973812\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−10.08072534316222640609880179955, −8.960320790585387478147860975644, −8.166076802972896131660435360600, −7.06424021617212550510788810925, −6.39054737840711598284307932890, −5.42145831975423179442916360935, −4.78434526310804408668386291359, −3.89424117562758728523112409100, −2.42537307729511615919253401926, −1.31389015134512166909255491495, 1.31389015134512166909255491495, 2.42537307729511615919253401926, 3.89424117562758728523112409100, 4.78434526310804408668386291359, 5.42145831975423179442916360935, 6.39054737840711598284307932890, 7.06424021617212550510788810925, 8.166076802972896131660435360600, 8.960320790585387478147860975644, 10.08072534316222640609880179955

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