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Properties

Label 2-2667-2667.1403-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.792 + 0.610i$
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  + (0.698 − 0.715i)3-s + (0.955 + 0.294i)4-s + (0.939 − 0.342i)7-s + (−0.0249 − 0.999i)9-s + (0.878 − 0.478i)12-s + (−1.42 + 0.178i)13-s + (0.826 + 0.563i)16-s + 0.396i·19-s + (0.411 − 0.911i)21-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)27-s + (0.998 − 0.0498i)28-s + (1.36 − 0.881i)31-s + (0.270 − 0.962i)36-s + (−0.233 − 0.0850i)37-s + ⋯
L(s)  = 1  + (0.698 − 0.715i)3-s + (0.955 + 0.294i)4-s + (0.939 − 0.342i)7-s + (−0.0249 − 0.999i)9-s + (0.878 − 0.478i)12-s + (−1.42 + 0.178i)13-s + (0.826 + 0.563i)16-s + 0.396i·19-s + (0.411 − 0.911i)21-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)27-s + (0.998 − 0.0498i)28-s + (1.36 − 0.881i)31-s + (0.270 − 0.962i)36-s + (−0.233 − 0.0850i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.792 + 0.610i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.792 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.030273813\)
\(L(\frac12)\) \(\approx\) \(2.030273813\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.698 + 0.715i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
127 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-0.921 - 0.388i)T^{2} \)
13 \( 1 + (1.42 - 0.178i)T + (0.969 - 0.246i)T^{2} \)
17 \( 1 + (-0.411 - 0.911i)T^{2} \)
19 \( 1 - 0.396iT - T^{2} \)
23 \( 1 + (0.921 - 0.388i)T^{2} \)
29 \( 1 + (0.980 - 0.198i)T^{2} \)
31 \( 1 + (-1.36 + 0.881i)T + (0.411 - 0.911i)T^{2} \)
37 \( 1 + (0.233 + 0.0850i)T + (0.766 + 0.642i)T^{2} \)
41 \( 1 + (-0.583 + 0.811i)T^{2} \)
43 \( 1 + (1.69 + 0.765i)T + (0.661 + 0.749i)T^{2} \)
47 \( 1 + (0.826 + 0.563i)T^{2} \)
53 \( 1 + (0.542 + 0.840i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (1.77 + 0.133i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (0.486 - 1.90i)T + (-0.878 - 0.478i)T^{2} \)
71 \( 1 + (0.124 - 0.992i)T^{2} \)
73 \( 1 + (0.970 + 0.381i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (-1.01 - 1.14i)T + (-0.124 + 0.992i)T^{2} \)
83 \( 1 + (-0.456 + 0.889i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
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

−8.688395012938987782732820969989, −8.015114411516055585064178550911, −7.42948630823244249675134446893, −6.99210801226943403844041672755, −6.09126865486308656024469276600, −5.06320744580817620968598678680, −4.02826111191419947897945988857, −3.01522593731350564618895469129, −2.20932759572142831591338262657, −1.41599846244047293010408430807, 1.68572432552879886900597161504, 2.54484814207832217106285166389, 3.17777538125839302796197971930, 4.76446464118983685983613452834, 4.82942803997176693389778257758, 6.01191874549240053023791515056, 6.95002536887031555904524039688, 7.76580838807741307365869280123, 8.245719310549694442756928031229, 9.137472389898577422895250279374

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