Properties

Label 2-2166-1.1-c1-0-27
Degree $2$
Conductor $2166$
Sign $1$
Analytic cond. $17.2955$
Root an. cond. $4.15879$
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 + 6-s + 7-s + 8-s + 9-s − 2·11-s + 12-s + 3·13-s + 14-s + 16-s + 4·17-s + 18-s + 21-s − 2·22-s + 4·23-s + 24-s − 5·25-s + 3·26-s + 27-s + 28-s + 3·31-s + 32-s − 2·33-s + 4·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.218·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s − 25-s + 0.588·26-s + 0.192·27-s + 0.188·28-s + 0.538·31-s + 0.176·32-s − 0.348·33-s + 0.685·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2166\)    =    \(2 \cdot 3 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(17.2955\)
Root analytic conductor: \(4.15879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2166} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2166,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.799455844\)
\(L(\frac12)\) \(\approx\) \(3.799455844\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−8.925966099863985529054739651698, −8.180578212475829505283816182977, −7.59187105262908473388016033500, −6.71460371174360847262151229869, −5.76177764095449448246208886611, −5.08258719095464042056098916585, −4.08698162864034153602834325880, −3.32245950530557880865147334790, −2.41032538782090925484227896417, −1.24755944995887851640988741563, 1.24755944995887851640988741563, 2.41032538782090925484227896417, 3.32245950530557880865147334790, 4.08698162864034153602834325880, 5.08258719095464042056098916585, 5.76177764095449448246208886611, 6.71460371174360847262151229869, 7.59187105262908473388016033500, 8.180578212475829505283816182977, 8.925966099863985529054739651698

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