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Properties

Label 2-2151-2151.238-c0-0-0
Degree $2$
Conductor $2151$
Sign $0.990 + 0.139i$
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  + (−0.961 − 1.66i)2-s + (−0.104 − 0.994i)3-s + (−1.34 + 2.33i)4-s + (−0.438 + 0.759i)5-s + (−1.55 + 1.13i)6-s + 3.26·8-s + (−0.978 + 0.207i)9-s + 1.68·10-s + (−0.990 − 1.71i)11-s + (2.46 + 1.09i)12-s + (0.800 + 0.356i)15-s + (−1.78 − 3.09i)16-s − 1.87·17-s + (1.28 + 1.42i)18-s + (−1.18 − 2.04i)20-s + ⋯
L(s)  = 1  + (−0.961 − 1.66i)2-s + (−0.104 − 0.994i)3-s + (−1.34 + 2.33i)4-s + (−0.438 + 0.759i)5-s + (−1.55 + 1.13i)6-s + 3.26·8-s + (−0.978 + 0.207i)9-s + 1.68·10-s + (−0.990 − 1.71i)11-s + (2.46 + 1.09i)12-s + (0.800 + 0.356i)15-s + (−1.78 − 3.09i)16-s − 1.87·17-s + (1.28 + 1.42i)18-s + (−1.18 − 2.04i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $0.990 + 0.139i$
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: \((2,\ 2151,\ (\ :0),\ 0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1704492191\)
\(L(\frac12)\) \(\approx\) \(0.1704492191\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 + 0.994i)T \)
239 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.961 + 1.66i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.438 - 0.759i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.990 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.882 - 1.52i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.241 + 0.419i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 0.618T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.0348 + 0.0604i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.039064092957533087123717493853, −8.643976601627250785659381694595, −7.995206573529105871234043295898, −7.23849084621625639549078917358, −6.40331433064993475811849645689, −5.10243373112189509198409932869, −3.82329723836560057989032189043, −2.85854797148097530182052609742, −2.54806737101904334721210422867, −1.14609242471702301432258480084, 0.18766786233196130148463484824, 2.19447773859883448727883555692, 4.31467145268230065303840776037, 4.66793727470845994369055037212, 5.23252386997935718963628145114, 6.34642199863760987467450637038, 6.96149404403434838466293422353, 8.032639000261438287228821045459, 8.335299227746792774496453959387, 9.192439069332688910070402707867

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