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Properties

Label 2-2667-2667.1049-c0-0-1
Degree $2$
Conductor $2667$
Sign $0.979 - 0.202i$
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  + (0.988 + 0.149i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)7-s + (0.955 + 0.294i)9-s + (−0.955 + 0.294i)12-s + (−0.297 + 0.0222i)13-s + (0.623 − 0.781i)16-s + 1.36i·19-s + (0.5 − 0.866i)21-s + (0.623 + 0.781i)25-s + (0.900 + 0.433i)27-s + (0.0747 + 0.997i)28-s + (1.04 − 1.53i)31-s + (−0.988 + 0.149i)36-s + (0.0747 − 0.129i)37-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)7-s + (0.955 + 0.294i)9-s + (−0.955 + 0.294i)12-s + (−0.297 + 0.0222i)13-s + (0.623 − 0.781i)16-s + 1.36i·19-s + (0.5 − 0.866i)21-s + (0.623 + 0.781i)25-s + (0.900 + 0.433i)27-s + (0.0747 + 0.997i)28-s + (1.04 − 1.53i)31-s + (−0.988 + 0.149i)36-s + (0.0747 − 0.129i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.979 - 0.202i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.979 - 0.202i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.514322412\)
\(L(\frac12)\) \(\approx\) \(1.514322412\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.988 - 0.149i)T \)
7 \( 1 + (-0.365 + 0.930i)T \)
127 \( 1 + T \)
good2 \( 1 + (0.900 - 0.433i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.297 - 0.0222i)T + (0.988 - 0.149i)T^{2} \)
17 \( 1 + (0.365 - 0.930i)T^{2} \)
19 \( 1 - 1.36iT - T^{2} \)
23 \( 1 + (0.0747 - 0.997i)T^{2} \)
29 \( 1 + (-0.733 + 0.680i)T^{2} \)
31 \( 1 + (-1.04 + 1.53i)T + (-0.365 - 0.930i)T^{2} \)
37 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.365 - 0.930i)T^{2} \)
43 \( 1 + (-1.81 + 0.712i)T + (0.733 - 0.680i)T^{2} \)
47 \( 1 + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.880 + 0.702i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.258 - 1.71i)T + (-0.955 - 0.294i)T^{2} \)
71 \( 1 + (-0.0747 + 0.997i)T^{2} \)
73 \( 1 + (0.574 + 0.131i)T + (0.900 + 0.433i)T^{2} \)
79 \( 1 + (0.733 - 0.680i)T + (0.0747 - 0.997i)T^{2} \)
83 \( 1 + (-0.826 - 0.563i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (-0.455 + 1.16i)T + (-0.733 - 0.680i)T^{2} \)
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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

−8.975631382851549670316846985877, −8.294149307026530821648542992549, −7.64421357638971495459007796134, −7.23287843798146578358634197371, −5.90569436191327359657978185253, −4.82730784573769147308828619382, −4.14057742023290461854677065060, −3.61989820746743735978721540511, −2.56079764389222066480518240416, −1.21387883036401944299960905382, 1.19775598902225158805937307409, 2.44498810228047256282476891554, 3.15768285856561506573994073492, 4.52656348068360477670179015895, 4.78072145980563425173031636232, 5.91296342880787866699064391247, 6.76001886640024688752630481808, 7.72690293908265479797111700165, 8.489341671959284814184117078387, 8.954820056518150760101327866639

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