Properties

Label 2-2151-2151.1672-c0-0-11
Degree $2$
Conductor $2151$
Sign $-0.719 + 0.694i$
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  + (−0.0348 + 0.0604i)2-s + (−0.978 + 0.207i)3-s + (0.497 + 0.861i)4-s + (−0.990 − 1.71i)5-s + (0.0215 − 0.0663i)6-s − 0.139·8-s + (0.913 − 0.406i)9-s + 0.138·10-s + (0.719 − 1.24i)11-s + (−0.665 − 0.739i)12-s + (1.32 + 1.47i)15-s + (−0.492 + 0.853i)16-s − 1.87·17-s + (−0.00729 + 0.0694i)18-s + (0.985 − 1.70i)20-s + ⋯
L(s)  = 1  + (−0.0348 + 0.0604i)2-s + (−0.978 + 0.207i)3-s + (0.497 + 0.861i)4-s + (−0.990 − 1.71i)5-s + (0.0215 − 0.0663i)6-s − 0.139·8-s + (0.913 − 0.406i)9-s + 0.138·10-s + (0.719 − 1.24i)11-s + (−0.665 − 0.739i)12-s + (1.32 + 1.47i)15-s + (−0.492 + 0.853i)16-s − 1.87·17-s + (−0.00729 + 0.0694i)18-s + (0.985 − 1.70i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(1.07348\)
Root analytic conductor: \(1.03609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1672, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :0),\ -0.719 + 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3635677863\)
\(L(\frac12)\) \(\approx\) \(0.3635677863\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.978 - 0.207i)T \)
239 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.0348 - 0.0604i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.990 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.719 + 1.24i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.438 - 0.759i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.848 + 1.46i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.559 - 0.968i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.890867434972306171871548107496, −8.295088386593369072238839474520, −7.45823625286447989755355179309, −6.62421769297424436946216789425, −5.79611828564500753491269270045, −4.84509145016947842561301962927, −4.09640189821687275279026649849, −3.57039449070937769263775137869, −1.72800265468679019486452856367, −0.28115858462310098589860699287, 1.75073112416500490422645763494, 2.64564685371062760462479795320, 4.04296158133291778906705894256, 4.65791611390124485192099964518, 5.93988331705236728173273893739, 6.56900464484773329180755923881, 7.12418721688795475856344485656, 7.40868470495312664501117803406, 8.892784635419936656203539894540, 9.975827051275844042664230487960

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