Properties

Label 2-2151-2151.238-c0-0-2
Degree $2$
Conductor $2151$
Sign $0.173 + 0.984i$
Analytic cond. $1.07348$
Root an. cond. $1.03609$
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  + (0.939 + 1.62i)2-s + (−0.5 + 0.866i)3-s + (−1.26 + 2.19i)4-s + (−0.173 + 0.300i)5-s − 1.87·6-s − 2.87·8-s + (−0.499 − 0.866i)9-s − 0.652·10-s + (−0.173 − 0.300i)11-s + (−1.26 − 2.19i)12-s + (−0.173 − 0.300i)15-s + (−1.43 − 2.49i)16-s − 1.87·17-s + (0.939 − 1.62i)18-s + (−0.439 − 0.761i)20-s + ⋯
L(s)  = 1  + (0.939 + 1.62i)2-s + (−0.5 + 0.866i)3-s + (−1.26 + 2.19i)4-s + (−0.173 + 0.300i)5-s − 1.87·6-s − 2.87·8-s + (−0.499 − 0.866i)9-s − 0.652·10-s + (−0.173 − 0.300i)11-s + (−1.26 − 2.19i)12-s + (−0.173 − 0.300i)15-s + (−1.43 − 2.49i)16-s − 1.87·17-s + (0.939 − 1.62i)18-s + (−0.439 − 0.761i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8885387204\)
\(L(\frac12)\) \(\approx\) \(0.8885387204\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.87T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.533888194589612177974357397157, −8.835634191427802305874090182157, −8.284401951653896459422364074392, −7.08928231482185620045690041881, −6.74056840997520500573970417614, −5.84369756202022242896691983830, −5.20225252725517574738031374859, −4.44879429440895800855772022491, −3.78626034075234929663284957098, −2.86253975413918395600563704371, 0.46456603179146260202630408286, 1.90619037866651601221210694555, 2.38668220169123174547162429190, 3.64457234661211147124053425892, 4.61893615851963534583311775643, 5.10563675585137393518917705686, 6.12350796761916419612689115212, 6.78742262621421082016946782281, 7.992371777479416445441818819419, 8.905093284009209226299050422708

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