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Properties

Label 2-2667-2667.1238-c0-0-1
Degree $2$
Conductor $2667$
Sign $-0.469 + 0.883i$
Analytic cond. $1.33100$
Root an. cond. $1.15369$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)7-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)12-s + (0.846 + 0.193i)13-s + (−0.900 + 0.433i)16-s − 1.56i·19-s − 21-s + (−0.900 − 0.433i)25-s + (0.222 − 0.974i)27-s + (−0.222 + 0.974i)28-s + (−0.846 + 0.193i)31-s + (−0.900 − 0.433i)36-s + 0.445·37-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)7-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)12-s + (0.846 + 0.193i)13-s + (−0.900 + 0.433i)16-s − 1.56i·19-s − 21-s + (−0.900 − 0.433i)25-s + (0.222 − 0.974i)27-s + (−0.222 + 0.974i)28-s + (−0.846 + 0.193i)31-s + (−0.900 − 0.433i)36-s + 0.445·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.469 + 0.883i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ -0.469 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.353380930\)
\(L(\frac12)\) \(\approx\) \(1.353380930\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
127 \( 1 + T \)
good2 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (-0.846 - 0.193i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-0.900 - 0.433i)T^{2} \)
19 \( 1 + 1.56iT - T^{2} \)
23 \( 1 + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.846 - 0.193i)T + (0.900 - 0.433i)T^{2} \)
37 \( 1 - 0.445T + T^{2} \)
41 \( 1 + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (-0.678 - 1.40i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.22 + 0.974i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 + (1.24 - 1.56i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
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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

−9.039917072241369426657303101623, −8.074084648737377969836917529328, −7.18825391560556157918085185079, −6.48783255090215605585629507070, −5.97877859529451403169538089340, −4.73050863803477777910597353559, −3.93003400517411746185543506157, −3.00575391011300814830330442391, −1.96614885647329457366123451961, −0.789656553900927581710481282457, 1.93296449299934845058115657757, 2.95054719465670724937203471936, 3.70401233783725913946838434443, 4.07223600158927117567227239466, 5.42038657920433887578729780929, 6.19734847256463517760732187770, 7.35137176192763832676095701735, 7.80319767590875065627627284295, 8.716808351910067638237927664521, 9.021627388627334294270795032524

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