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Properties

Label 2-2667-2667.1010-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.470 + 0.882i$
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  + (0.270 + 0.962i)3-s + (−0.733 + 0.680i)4-s + (−0.411 − 0.911i)7-s + (−0.853 + 0.521i)9-s + (−0.853 − 0.521i)12-s + (−0.173 − 1.38i)13-s + (0.0747 − 0.997i)16-s + (−0.980 − 1.69i)19-s + (0.766 − 0.642i)21-s + (−0.900 − 0.433i)25-s + (−0.733 − 0.680i)27-s + (0.921 + 0.388i)28-s + (1.16 − 0.0581i)31-s + (0.270 − 0.962i)36-s + (−0.0431 − 0.244i)37-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)3-s + (−0.733 + 0.680i)4-s + (−0.411 − 0.911i)7-s + (−0.853 + 0.521i)9-s + (−0.853 − 0.521i)12-s + (−0.173 − 1.38i)13-s + (0.0747 − 0.997i)16-s + (−0.980 − 1.69i)19-s + (0.766 − 0.642i)21-s + (−0.900 − 0.433i)25-s + (−0.733 − 0.680i)27-s + (0.921 + 0.388i)28-s + (1.16 − 0.0581i)31-s + (0.270 − 0.962i)36-s + (−0.0431 − 0.244i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.470 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5811305226\)
\(L(\frac12)\) \(\approx\) \(0.5811305226\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.270 - 0.962i)T \)
7 \( 1 + (0.411 + 0.911i)T \)
127 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.733 - 0.680i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.124 + 0.992i)T^{2} \)
13 \( 1 + (0.173 + 1.38i)T + (-0.969 + 0.246i)T^{2} \)
17 \( 1 + (0.583 - 0.811i)T^{2} \)
19 \( 1 + (0.980 + 1.69i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.124 - 0.992i)T^{2} \)
29 \( 1 + (-0.980 + 0.198i)T^{2} \)
31 \( 1 + (-1.16 + 0.0581i)T + (0.995 - 0.0995i)T^{2} \)
37 \( 1 + (0.0431 + 0.244i)T + (-0.939 + 0.342i)T^{2} \)
41 \( 1 + (0.583 - 0.811i)T^{2} \)
43 \( 1 + (0.300 - 0.666i)T + (-0.661 - 0.749i)T^{2} \)
47 \( 1 + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.456 + 0.889i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.822 - 0.395i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 + (-0.242 + 0.248i)T + (-0.0249 - 0.999i)T^{2} \)
71 \( 1 + (0.124 - 0.992i)T^{2} \)
73 \( 1 + (1.06 + 1.33i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.487 - 1.45i)T + (-0.797 - 0.603i)T^{2} \)
83 \( 1 + (-0.456 + 0.889i)T^{2} \)
89 \( 1 + (-0.955 - 0.294i)T^{2} \)
97 \( 1 + (0.0872 - 0.121i)T + (-0.318 - 0.947i)T^{2} \)
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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

−8.940011161090373236661113339085, −8.185275423817794718353663249715, −7.68383385219692370393717261721, −6.67128579134216218851849153620, −5.61538889908833413884921037533, −4.62594988251380555601388322442, −4.26659355789741704729250361957, −3.25704310741790114218601200481, −2.68624208460391656685590858598, −0.36223513579917873056480391643, 1.55121692120778891524627540939, 2.21922306770972124569599311856, 3.52881913681497875308013614403, 4.42778757554616554184743928048, 5.53892461464073382972332050810, 6.15496804201737881879118779769, 6.65767553930335582167357089951, 7.78678544250587408397669988860, 8.534998277215652598463666454133, 9.019681879874627735502794202637

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