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Properties

Label 2-2667-2667.1328-c0-0-0
Degree $2$
Conductor $2667$
Sign $-0.104 + 0.994i$
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  + (−0.583 − 0.811i)3-s + (−0.733 + 0.680i)4-s + (−0.173 − 0.984i)7-s + (−0.318 + 0.947i)9-s + (0.980 + 0.198i)12-s + (−0.0809 + 1.62i)13-s + (0.0747 − 0.997i)16-s − 1.04i·19-s + (−0.698 + 0.715i)21-s + (0.826 − 0.563i)25-s + (0.955 − 0.294i)27-s + (0.797 + 0.603i)28-s + (−0.191 − 0.454i)31-s + (−0.411 − 0.911i)36-s + (0.346 − 1.96i)37-s + ⋯
L(s)  = 1  + (−0.583 − 0.811i)3-s + (−0.733 + 0.680i)4-s + (−0.173 − 0.984i)7-s + (−0.318 + 0.947i)9-s + (0.980 + 0.198i)12-s + (−0.0809 + 1.62i)13-s + (0.0747 − 0.997i)16-s − 1.04i·19-s + (−0.698 + 0.715i)21-s + (0.826 − 0.563i)25-s + (0.955 − 0.294i)27-s + (0.797 + 0.603i)28-s + (−0.191 − 0.454i)31-s + (−0.411 − 0.911i)36-s + (0.346 − 1.96i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.104 + 0.994i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ -0.104 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6155165191\)
\(L(\frac12)\) \(\approx\) \(0.6155165191\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.583 + 0.811i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
127 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.733 - 0.680i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.456 - 0.889i)T^{2} \)
13 \( 1 + (0.0809 - 1.62i)T + (-0.995 - 0.0995i)T^{2} \)
17 \( 1 + (0.698 + 0.715i)T^{2} \)
19 \( 1 + 1.04iT - T^{2} \)
23 \( 1 + (0.456 - 0.889i)T^{2} \)
29 \( 1 + (-0.853 + 0.521i)T^{2} \)
31 \( 1 + (0.191 + 0.454i)T + (-0.698 + 0.715i)T^{2} \)
37 \( 1 + (-0.346 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
41 \( 1 + (-0.969 + 0.246i)T^{2} \)
43 \( 1 + (0.213 + 0.208i)T + (0.0249 + 0.999i)T^{2} \)
47 \( 1 + (0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.921 - 0.388i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (1.11 - 1.63i)T + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.128 + 1.27i)T + (-0.980 + 0.198i)T^{2} \)
71 \( 1 + (0.998 - 0.0498i)T^{2} \)
73 \( 1 + (0.223 + 1.48i)T + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.0468 + 1.87i)T + (-0.998 + 0.0498i)T^{2} \)
83 \( 1 + (0.124 - 0.992i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-1.60 + 0.407i)T + (0.878 - 0.478i)T^{2} \)
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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

−8.915624330834442909280798814960, −7.88041009176911011610274853729, −7.23092920308501816874126715422, −6.80166161973981960888956970953, −5.84595197869647787913221195283, −4.60884391247075747003552996222, −4.36870427465280077670649236532, −3.13191584107165325603463134153, −1.94729245670406392623368569603, −0.50317361644223227218192068102, 1.19633186681374278428653025434, 2.89004548973683278419367105940, 3.67194126213470987266600932776, 4.83786816726228293979713494412, 5.30276268459886455927054957826, 5.89915763372497356816659842184, 6.60274483910987723379830134599, 8.051586839949211618079992684196, 8.581426628272796228146686657050, 9.403970993569935208575696060099

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