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Properties

Label 2-2667-2667.1475-c0-0-0
Degree $2$
Conductor $2667$
Sign $0.845 - 0.534i$
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  + (0.878 + 0.478i)3-s + (0.826 + 0.563i)4-s + (−0.173 − 0.984i)7-s + (0.542 + 0.840i)9-s + (0.456 + 0.889i)12-s + (0.920 + 0.259i)13-s + (0.365 + 0.930i)16-s − 1.98i·19-s + (0.318 − 0.947i)21-s + (−0.988 + 0.149i)25-s + (0.0747 + 0.997i)27-s + (0.411 − 0.911i)28-s + (−0.321 + 0.231i)31-s + (−0.0249 + 0.999i)36-s + (−0.0940 + 0.533i)37-s + ⋯
L(s)  = 1  + (0.878 + 0.478i)3-s + (0.826 + 0.563i)4-s + (−0.173 − 0.984i)7-s + (0.542 + 0.840i)9-s + (0.456 + 0.889i)12-s + (0.920 + 0.259i)13-s + (0.365 + 0.930i)16-s − 1.98i·19-s + (0.318 − 0.947i)21-s + (−0.988 + 0.149i)25-s + (0.0747 + 0.997i)27-s + (0.411 − 0.911i)28-s + (−0.321 + 0.231i)31-s + (−0.0249 + 0.999i)36-s + (−0.0940 + 0.533i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $0.845 - 0.534i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ 0.845 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.029965956\)
\(L(\frac12)\) \(\approx\) \(2.029965956\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.878 - 0.478i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
127 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.826 - 0.563i)T^{2} \)
5 \( 1 + (0.988 - 0.149i)T^{2} \)
11 \( 1 + (0.969 - 0.246i)T^{2} \)
13 \( 1 + (-0.920 - 0.259i)T + (0.853 + 0.521i)T^{2} \)
17 \( 1 + (-0.318 - 0.947i)T^{2} \)
19 \( 1 + 1.98iT - T^{2} \)
23 \( 1 + (-0.969 - 0.246i)T^{2} \)
29 \( 1 + (-0.124 + 0.992i)T^{2} \)
31 \( 1 + (0.321 - 0.231i)T + (0.318 - 0.947i)T^{2} \)
37 \( 1 + (0.0940 - 0.533i)T + (-0.939 - 0.342i)T^{2} \)
41 \( 1 + (0.980 + 0.198i)T^{2} \)
43 \( 1 + (1.28 + 0.433i)T + (0.797 + 0.603i)T^{2} \)
47 \( 1 + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (-0.583 - 0.811i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.0296 - 0.196i)T + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (-0.670 - 1.09i)T + (-0.456 + 0.889i)T^{2} \)
71 \( 1 + (-0.270 - 0.962i)T^{2} \)
73 \( 1 + (0.0678 - 0.0730i)T + (-0.0747 - 0.997i)T^{2} \)
79 \( 1 + (1.49 + 1.13i)T + (0.270 + 0.962i)T^{2} \)
83 \( 1 + (-0.995 + 0.0995i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
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

−8.937880370774214812584418905304, −8.406894612337761803201331514351, −7.51718221240513404715348624520, −7.03912590196848984383309782357, −6.29406741984943883258610740088, −4.98977425407113697996389189125, −4.07166207382299882236095957889, −3.45961070737179156832556784431, −2.64582580779389253581348183025, −1.55668257187491016988606215205, 1.49437874672220028709620626728, 2.12112196933702095993521660425, 3.15569059744236531757655713864, 3.86696062819479341499261505399, 5.38221967881106549991733068845, 6.06435380115587847293822176438, 6.52584333703683460556569382392, 7.62928214968060946218932401685, 8.125076400379406822719290532133, 8.874957699754310257874063372902

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