Properties

Label 2-2888-8.3-c0-0-1
Degree $2$
Conductor $2888$
Sign $1$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.347·3-s + 4-s + 0.347·6-s − 8-s − 0.879·9-s + 1.53·11-s − 0.347·12-s + 16-s − 17-s + 0.879·18-s − 1.53·22-s + 0.347·24-s + 25-s + 0.652·27-s − 32-s − 0.532·33-s + 34-s − 0.879·36-s + 1.87·41-s − 43-s + 1.53·44-s − 0.347·48-s + 49-s − 50-s + 0.347·51-s − 0.652·54-s + ⋯
L(s)  = 1  − 2-s − 0.347·3-s + 4-s + 0.347·6-s − 8-s − 0.879·9-s + 1.53·11-s − 0.347·12-s + 16-s − 17-s + 0.879·18-s − 1.53·22-s + 0.347·24-s + 25-s + 0.652·27-s − 32-s − 0.532·33-s + 34-s − 0.879·36-s + 1.87·41-s − 43-s + 1.53·44-s − 0.347·48-s + 49-s − 50-s + 0.347·51-s − 0.652·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (723, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6881023955\)
\(L(\frac12)\) \(\approx\) \(0.6881023955\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + 0.347T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.53T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.87T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.53T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.87T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.53T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.87T + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - 1.87T + T^{2} \)
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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

−8.993678517698826143931529905819, −8.429250142831578527476831727452, −7.48526290844213018085038823308, −6.57370698852761472192408807411, −6.30386950254554263872415327435, −5.28674696023220523436484581613, −4.16207856650570433711671946561, −3.11878518588072946754418955709, −2.12604960266300348604347603922, −0.898999948754934995127462865547, 0.898999948754934995127462865547, 2.12604960266300348604347603922, 3.11878518588072946754418955709, 4.16207856650570433711671946561, 5.28674696023220523436484581613, 6.30386950254554263872415327435, 6.57370698852761472192408807411, 7.48526290844213018085038823308, 8.429250142831578527476831727452, 8.993678517698826143931529905819

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