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Properties

Label 2-2667-2667.1070-c0-0-1
Degree $2$
Conductor $2667$
Sign $0.870 + 0.491i$
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  + (0.955 + 0.294i)3-s + (0.623 − 0.781i)4-s + (0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)12-s + (−0.0878 − 0.582i)13-s + (−0.222 − 0.974i)16-s + 1.99i·19-s + (0.733 − 0.680i)21-s + (−0.222 + 0.974i)25-s + (0.623 + 0.781i)27-s + (−0.365 − 0.930i)28-s + (−1.26 + 0.496i)31-s + (0.955 − 0.294i)36-s + (−0.988 − 1.71i)37-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)3-s + (0.623 − 0.781i)4-s + (0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)12-s + (−0.0878 − 0.582i)13-s + (−0.222 − 0.974i)16-s + 1.99i·19-s + (0.733 − 0.680i)21-s + (−0.222 + 0.974i)25-s + (0.623 + 0.781i)27-s + (−0.365 − 0.930i)28-s + (−1.26 + 0.496i)31-s + (0.955 − 0.294i)36-s + (−0.988 − 1.71i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.870 + 0.491i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1070, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.870 + 0.491i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.059647182\)
\(L(\frac12)\) \(\approx\) \(2.059647182\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.955 - 0.294i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
127 \( 1 + T \)
good2 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.0878 + 0.582i)T + (-0.955 + 0.294i)T^{2} \)
17 \( 1 + (-0.733 - 0.680i)T^{2} \)
19 \( 1 - 1.99iT - T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T^{2} \)
29 \( 1 + (0.0747 - 0.997i)T^{2} \)
31 \( 1 + (1.26 - 0.496i)T + (0.733 - 0.680i)T^{2} \)
37 \( 1 + (0.988 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.733 - 0.680i)T^{2} \)
43 \( 1 + (0.590 + 0.636i)T + (-0.0747 + 0.997i)T^{2} \)
47 \( 1 + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.365 + 0.930i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.81 + 0.414i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (0.510 - 1.65i)T + (-0.826 - 0.563i)T^{2} \)
71 \( 1 + (0.988 + 0.149i)T^{2} \)
73 \( 1 + (0.488 - 1.01i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.0747 + 0.997i)T + (-0.988 - 0.149i)T^{2} \)
83 \( 1 + (-0.365 - 0.930i)T^{2} \)
89 \( 1 + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.326 + 0.302i)T + (0.0747 + 0.997i)T^{2} \)
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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

−8.982611230243183867625880843380, −8.156890613850077144131525066144, −7.40820685564210082474645979564, −7.01488597732158010651436069905, −5.69727410682954170975242132664, −5.19806598471911555349919707404, −3.96242099597276756695158081145, −3.42476600940953787754787573145, −2.09383628789787844098234379281, −1.41430514379260358288165571681, 1.75623126789353345235107100646, 2.48345496599041085210459072564, 3.16856046226455469895316787952, 4.22507211035848562099348056333, 5.05552570692376186974154398186, 6.40033906874722124501560455306, 6.87451630518471382518172220879, 7.69024490870102797351443807699, 8.362558857964214397136789034395, 8.902949309770719563166837956431

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