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Properties

Label 2-2667-2667.527-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.955 + 0.295i$
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  + (−0.866 − 0.5i)2-s + 3-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + 7-s + i·8-s + 9-s + (−0.499 − 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + 3-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + 7-s + i·8-s + 9-s + (−0.499 − 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + (−0.866 − 0.5i)14-s + (0.866 + 0.5i)15-s + (0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.955 + 0.295i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.955 + 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394347113\)
\(L(\frac12)\) \(\approx\) \(1.394347113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.213288874009730666990185206698, −8.552360940602064423181074646255, −7.59817372052423230328245700916, −7.03724468750631782763217133576, −6.04260376681931616451069005434, −4.85129773766684924630586651026, −4.29038098059449243349544037908, −2.83259327979317011591102733342, −1.99374414837342553323313429235, −1.58970722134816728131174151537, 1.29636304386369073798428479953, 2.07736244357152833921393578238, 3.47419586586901136387647998058, 4.19653641881001441799985094758, 5.26420220560288900879555078588, 6.15581031409663268996151897013, 7.14102718098616900150361340374, 7.78381466511821579876118229673, 8.439747826430203856582364281706, 8.933584602214892236591991540645

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