Properties

Label 2-1110-1.1-c1-0-13
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 + 4·7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 2·13-s + 4·14-s − 15-s + 16-s − 2·17-s + 18-s + 2·19-s − 20-s + 4·21-s − 2·22-s + 24-s + 25-s + 2·26-s + 27-s + 4·28-s + 2·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.872·21-s − 0.426·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.755·28-s + 0.371·29-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

\(L(1)\) \(\approx\) \(3.323028531\)
\(L(\frac12)\) \(\approx\) \(3.323028531\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−9.997711853294076131540262050453, −8.678448890720004138812556129253, −8.175086931274304777032749837299, −7.48044158538181617965029617091, −6.51010129479966833930515773997, −5.25256359120867251767225408448, −4.65757573905330316522778821333, −3.70994421105600557462843234506, −2.60721282477734981997386477165, −1.46077007465007974786279625764, 1.46077007465007974786279625764, 2.60721282477734981997386477165, 3.70994421105600557462843234506, 4.65757573905330316522778821333, 5.25256359120867251767225408448, 6.51010129479966833930515773997, 7.48044158538181617965029617091, 8.175086931274304777032749837299, 8.678448890720004138812556129253, 9.997711853294076131540262050453

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