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Properties

Label 2-2668-667.666-c0-0-0
Degree $2$
Conductor $2668$
Sign $1$
Analytic cond. $1.33150$
Root an. cond. $1.15390$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 2·11-s − 13-s + 17-s + 19-s + 23-s + 25-s + 29-s + 37-s + 43-s + 49-s − 59-s − 2·61-s − 71-s + 79-s + 81-s + 89-s − 2·97-s − 2·99-s − 2·113-s − 117-s + ⋯
L(s)  = 1  + 9-s − 2·11-s − 13-s + 17-s + 19-s + 23-s + 25-s + 29-s + 37-s + 43-s + 49-s − 59-s − 2·61-s − 71-s + 79-s + 81-s + 89-s − 2·97-s − 2·99-s − 2·113-s − 117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(1.33150\)
Root analytic conductor: \(1.15390\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2668} (1333, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2668,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.223742720\)
\(L(\frac12)\) \(\approx\) \(1.223742720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.185444854431975239965005995976, −8.001991569537845218827166890648, −7.56328025431698950990832840335, −7.01156509023048250602637000158, −5.80494941056169582017385001877, −5.03073546346907956902246888774, −4.55665282614296662153886085281, −3.11435773668680082704381426255, −2.57387461171942618544946486929, −1.07312048464250572711702473429, 1.07312048464250572711702473429, 2.57387461171942618544946486929, 3.11435773668680082704381426255, 4.55665282614296662153886085281, 5.03073546346907956902246888774, 5.80494941056169582017385001877, 7.01156509023048250602637000158, 7.56328025431698950990832840335, 8.001991569537845218827166890648, 9.185444854431975239965005995976

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