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Properties

Label 2-2667-2667.1130-c0-0-0
Degree $2$
Conductor $2667$
Sign $-0.510 - 0.859i$
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 + (0.955 + 0.294i)4-s + (−0.173 + 0.984i)7-s + (−0.853 + 0.521i)9-s + (−0.0249 + 0.999i)12-s + (−0.747 + 1.77i)13-s + (0.826 + 0.563i)16-s − 1.89i·19-s + (−0.995 + 0.0995i)21-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)27-s + (−0.456 + 0.889i)28-s + (−0.0908 − 1.82i)31-s + (−0.969 + 0.246i)36-s + (−0.320 − 1.81i)37-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)3-s + (0.955 + 0.294i)4-s + (−0.173 + 0.984i)7-s + (−0.853 + 0.521i)9-s + (−0.0249 + 0.999i)12-s + (−0.747 + 1.77i)13-s + (0.826 + 0.563i)16-s − 1.89i·19-s + (−0.995 + 0.0995i)21-s + (0.0747 + 0.997i)25-s + (−0.733 − 0.680i)27-s + (−0.456 + 0.889i)28-s + (−0.0908 − 1.82i)31-s + (−0.969 + 0.246i)36-s + (−0.320 − 1.81i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.510 - 0.859i$
Analytic conductor: \(1.33100\)
Root analytic conductor: \(1.15369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (1130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2667,\ (\ :0),\ -0.510 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.489644059\)
\(L(\frac12)\) \(\approx\) \(1.489644059\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T \)
127 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (0.797 - 0.603i)T^{2} \)
13 \( 1 + (0.747 - 1.77i)T + (-0.698 - 0.715i)T^{2} \)
17 \( 1 + (0.995 + 0.0995i)T^{2} \)
19 \( 1 + 1.89iT - T^{2} \)
23 \( 1 + (-0.797 - 0.603i)T^{2} \)
29 \( 1 + (-0.318 + 0.947i)T^{2} \)
31 \( 1 + (0.0908 + 1.82i)T + (-0.995 + 0.0995i)T^{2} \)
37 \( 1 + (0.320 + 1.81i)T + (-0.939 + 0.342i)T^{2} \)
41 \( 1 + (-0.411 - 0.911i)T^{2} \)
43 \( 1 + (-0.185 - 1.85i)T + (-0.980 + 0.198i)T^{2} \)
47 \( 1 + (0.826 + 0.563i)T^{2} \)
53 \( 1 + (-0.998 + 0.0498i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-1.67 - 0.125i)T + (0.988 + 0.149i)T^{2} \)
67 \( 1 + (-0.920 - 0.897i)T + (0.0249 + 0.999i)T^{2} \)
71 \( 1 + (-0.921 + 0.388i)T^{2} \)
73 \( 1 + (0.890 + 0.349i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (-1.84 + 0.372i)T + (0.921 - 0.388i)T^{2} \)
83 \( 1 + (-0.542 - 0.840i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.0614 - 0.136i)T + (-0.661 + 0.749i)T^{2} \)
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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

−9.298568003199308108981327910270, −8.798590599682012620815290321639, −7.74493153743412712248117690778, −7.01681545386549658288194082344, −6.23664691364842970462499098198, −5.36376516217890894385084349818, −4.53503805610697314098864990759, −3.64373329165152743319098355690, −2.52029870666090938881290323048, −2.20212976812988247447008573068, 0.892531483649218376880896555421, 1.90985345005853766136294144792, 2.99629560107977427493520609581, 3.58657878967351946440333687555, 5.14679233860172514782311622370, 5.83712977426780731466871173046, 6.72347670962667875780326995157, 7.14900132267396960764347394165, 8.119061430989129720153125019958, 8.206397964217556467426337541795

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