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Properties

Label 2-2667-2667.110-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.146 + 0.989i$
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  + (−0.411 + 0.911i)3-s + (−0.733 + 0.680i)4-s + (−0.766 + 0.642i)7-s + (−0.661 − 0.749i)9-s + (−0.318 − 0.947i)12-s + (−1.62 + 0.831i)13-s + (0.0747 − 0.997i)16-s − 0.956i·19-s + (−0.270 − 0.962i)21-s + (0.826 − 0.563i)25-s + (0.955 − 0.294i)27-s + (0.124 − 0.992i)28-s + (−0.864 + 1.14i)31-s + (0.995 + 0.0995i)36-s + (−0.698 − 0.586i)37-s + ⋯
L(s)  = 1  + (−0.411 + 0.911i)3-s + (−0.733 + 0.680i)4-s + (−0.766 + 0.642i)7-s + (−0.661 − 0.749i)9-s + (−0.318 − 0.947i)12-s + (−1.62 + 0.831i)13-s + (0.0747 − 0.997i)16-s − 0.956i·19-s + (−0.270 − 0.962i)21-s + (0.826 − 0.563i)25-s + (0.955 − 0.294i)27-s + (0.124 − 0.992i)28-s + (−0.864 + 1.14i)31-s + (0.995 + 0.0995i)36-s + (−0.698 − 0.586i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.146 + 0.989i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.146 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04663754886\)
\(L(\frac12)\) \(\approx\) \(0.04663754886\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.411 - 0.911i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
127 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.542 + 0.840i)T^{2} \)
13 \( 1 + (1.62 - 0.831i)T + (0.583 - 0.811i)T^{2} \)
17 \( 1 + (0.270 - 0.962i)T^{2} \)
19 \( 1 + 0.956iT - T^{2} \)
23 \( 1 + (0.542 + 0.840i)T^{2} \)
29 \( 1 + (0.878 + 0.478i)T^{2} \)
31 \( 1 + (0.864 - 1.14i)T + (-0.270 - 0.962i)T^{2} \)
37 \( 1 + (0.698 + 0.586i)T + (0.173 + 0.984i)T^{2} \)
41 \( 1 + (0.698 + 0.715i)T^{2} \)
43 \( 1 + (-0.286 + 0.0807i)T + (0.853 - 0.521i)T^{2} \)
47 \( 1 + (0.0747 - 0.997i)T^{2} \)
53 \( 1 + (-0.797 - 0.603i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.437 + 0.641i)T + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.555 - 0.399i)T + (0.318 - 0.947i)T^{2} \)
71 \( 1 + (-0.456 + 0.889i)T^{2} \)
73 \( 1 + (0.0590 + 0.391i)T + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (-0.296 + 0.181i)T + (0.456 - 0.889i)T^{2} \)
83 \( 1 + (-0.921 + 0.388i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (1.15 + 1.18i)T + (-0.0249 + 0.999i)T^{2} \)
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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

−9.130362281435943816740257617522, −8.412939443855736785041257648674, −7.18375680168961869166794940611, −6.68007970350437761281783131615, −5.44927351434668317494669646079, −4.91625402697021666629675584118, −4.19689311773865134436688835487, −3.23284430407149788656336346822, −2.50412716881009633418007744766, −0.03499118923626855400275214086, 1.18963419225398483535035233250, 2.45780186297237720885192702443, 3.58612414935001199828299238594, 4.70984724013351761245767294944, 5.43827243559811504355397492877, 6.06301679951223786978202242628, 6.99270898482599749029678119967, 7.54815545242864049222378190464, 8.340501623680047639148675044763, 9.331668142874094517529772490861

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