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Properties

Label 2-2667-2667.1454-c0-0-0
Degree $2$
Conductor $2667$
Sign $-0.568 - 0.822i$
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  + (0.0249 − 0.999i)3-s + (0.0747 + 0.997i)4-s + (−0.318 + 0.947i)7-s + (−0.998 − 0.0498i)9-s + (0.998 − 0.0498i)12-s + (−0.920 + 0.259i)13-s + (−0.988 + 0.149i)16-s + (−1.71 + 0.992i)19-s + (0.939 + 0.342i)21-s + (0.623 − 0.781i)25-s + (−0.0747 + 0.997i)27-s + (−0.969 − 0.246i)28-s + (−0.361 + 0.162i)31-s + (−0.0249 − 0.999i)36-s + (−0.414 + 0.348i)37-s + ⋯
L(s)  = 1  + (0.0249 − 0.999i)3-s + (0.0747 + 0.997i)4-s + (−0.318 + 0.947i)7-s + (−0.998 − 0.0498i)9-s + (0.998 − 0.0498i)12-s + (−0.920 + 0.259i)13-s + (−0.988 + 0.149i)16-s + (−1.71 + 0.992i)19-s + (0.939 + 0.342i)21-s + (0.623 − 0.781i)25-s + (−0.0747 + 0.997i)27-s + (−0.969 − 0.246i)28-s + (−0.361 + 0.162i)31-s + (−0.0249 − 0.999i)36-s + (−0.414 + 0.348i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.568 - 0.822i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1454, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ -0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5443803136\)
\(L(\frac12)\) \(\approx\) \(0.5443803136\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0249 + 0.999i)T \)
7 \( 1 + (0.318 - 0.947i)T \)
127 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.0747 - 0.997i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.270 - 0.962i)T^{2} \)
13 \( 1 + (0.920 - 0.259i)T + (0.853 - 0.521i)T^{2} \)
17 \( 1 + (0.980 - 0.198i)T^{2} \)
19 \( 1 + (1.71 - 0.992i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.270 - 0.962i)T^{2} \)
29 \( 1 + (-0.124 - 0.992i)T^{2} \)
31 \( 1 + (0.361 - 0.162i)T + (0.661 - 0.749i)T^{2} \)
37 \( 1 + (0.414 - 0.348i)T + (0.173 - 0.984i)T^{2} \)
41 \( 1 + (0.980 - 0.198i)T^{2} \)
43 \( 1 + (1.28 - 0.433i)T + (0.797 - 0.603i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.995 + 0.0995i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.155 + 0.124i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.614 - 1.12i)T + (-0.542 + 0.840i)T^{2} \)
71 \( 1 + (-0.270 + 0.962i)T^{2} \)
73 \( 1 + (0.0971 - 0.0221i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (-1.73 - 0.730i)T + (0.698 + 0.715i)T^{2} \)
83 \( 1 + (-0.995 - 0.0995i)T^{2} \)
89 \( 1 + (-0.826 + 0.563i)T^{2} \)
97 \( 1 + (1.93 - 0.391i)T + (0.921 - 0.388i)T^{2} \)
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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

−8.968851867267244714411206447502, −8.364476724289758175266684354403, −7.968889426653050470920158685698, −6.84457305896131762300298416196, −6.57424236662852062995427924938, −5.58089744027090499117405687460, −4.57015308421378607969474089876, −3.46101716148643551034009980559, −2.54625990856384573935314932025, −1.95674190505185690294153619518, 0.31504280582611521502677159392, 2.03718649417415261333573663747, 3.12391120429969277193868186842, 4.17649925754740779722400504089, 4.82425387694181953699561777108, 5.47101364561885729415874899829, 6.52584284001717070308054349967, 7.02019678467249120632547071946, 8.154314189475101233494915229949, 9.073267116898167098723303554699

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