Properties

Label 2-2151-1.1-c1-0-98
Degree $2$
Conductor $2151$
Sign $-1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.66·2-s + 5.08·4-s − 2.46·5-s − 5.08·7-s + 8.20·8-s − 6.55·10-s − 0.841·11-s − 4.31·13-s − 13.5·14-s + 11.6·16-s − 0.607·17-s − 7.12·19-s − 12.5·20-s − 2.24·22-s + 6.45·23-s + 1.05·25-s − 11.4·26-s − 25.8·28-s − 0.953·29-s + 1.79·31-s + 14.6·32-s − 1.61·34-s + 12.5·35-s − 4.38·37-s − 18.9·38-s − 20.1·40-s − 9.65·41-s + ⋯
L(s)  = 1  + 1.88·2-s + 2.54·4-s − 1.10·5-s − 1.92·7-s + 2.90·8-s − 2.07·10-s − 0.253·11-s − 1.19·13-s − 3.61·14-s + 2.91·16-s − 0.147·17-s − 1.63·19-s − 2.79·20-s − 0.477·22-s + 1.34·23-s + 0.211·25-s − 2.25·26-s − 4.88·28-s − 0.177·29-s + 0.322·31-s + 2.59·32-s − 0.277·34-s + 2.11·35-s − 0.720·37-s − 3.07·38-s − 3.19·40-s − 1.50·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $-1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 + T \)
good2 \( 1 - 2.66T + 2T^{2} \)
5 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 + 5.08T + 7T^{2} \)
11 \( 1 + 0.841T + 11T^{2} \)
13 \( 1 + 4.31T + 13T^{2} \)
17 \( 1 + 0.607T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 - 6.45T + 23T^{2} \)
29 \( 1 + 0.953T + 29T^{2} \)
31 \( 1 - 1.79T + 31T^{2} \)
37 \( 1 + 4.38T + 37T^{2} \)
41 \( 1 + 9.65T + 41T^{2} \)
43 \( 1 - 5.60T + 43T^{2} \)
47 \( 1 + 5.73T + 47T^{2} \)
53 \( 1 - 0.509T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 - 4.53T + 71T^{2} \)
73 \( 1 - 8.87T + 73T^{2} \)
79 \( 1 + 1.69T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−8.522379504925266162846037691732, −7.40245642306963201977355486264, −6.84685805956770603148025147947, −6.36340901171219824733559463294, −5.32794235704741450462463123742, −4.52168508138228043625031381166, −3.75064384160518059522643200868, −3.11445032287803013096827216672, −2.34944606133680269819496636157, 0, 2.34944606133680269819496636157, 3.11445032287803013096827216672, 3.75064384160518059522643200868, 4.52168508138228043625031381166, 5.32794235704741450462463123742, 6.36340901171219824733559463294, 6.84685805956770603148025147947, 7.40245642306963201977355486264, 8.522379504925266162846037691732

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