Properties

Label 2-2151-1.1-c1-0-76
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  − 0.104·2-s − 1.98·4-s + 0.431·5-s + 2.76·7-s + 0.415·8-s − 0.0449·10-s + 1.08·11-s − 2.81·13-s − 0.288·14-s + 3.93·16-s − 5.82·17-s − 7.25·19-s − 0.859·20-s − 0.113·22-s + 3.48·23-s − 4.81·25-s + 0.292·26-s − 5.50·28-s + 5.03·29-s − 2.02·31-s − 1.24·32-s + 0.606·34-s + 1.19·35-s + 2.08·37-s + 0.755·38-s + 0.179·40-s + 5.40·41-s + ⋯
L(s)  = 1  − 0.0736·2-s − 0.994·4-s + 0.193·5-s + 1.04·7-s + 0.146·8-s − 0.0142·10-s + 0.328·11-s − 0.779·13-s − 0.0769·14-s + 0.983·16-s − 1.41·17-s − 1.66·19-s − 0.192·20-s − 0.0241·22-s + 0.727·23-s − 0.962·25-s + 0.0574·26-s − 1.03·28-s + 0.935·29-s − 0.363·31-s − 0.219·32-s + 0.104·34-s + 0.202·35-s + 0.342·37-s + 0.122·38-s + 0.0283·40-s + 0.843·41-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 \( 1 + 0.104T + 2T^{2} \)
5 \( 1 - 0.431T + 5T^{2} \)
7 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 - 1.08T + 11T^{2} \)
13 \( 1 + 2.81T + 13T^{2} \)
17 \( 1 + 5.82T + 17T^{2} \)
19 \( 1 + 7.25T + 19T^{2} \)
23 \( 1 - 3.48T + 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 - 2.08T + 37T^{2} \)
41 \( 1 - 5.40T + 41T^{2} \)
43 \( 1 + 5.91T + 43T^{2} \)
47 \( 1 - 0.795T + 47T^{2} \)
53 \( 1 - 3.47T + 53T^{2} \)
59 \( 1 - 9.03T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 3.98T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 1.10T + 73T^{2} \)
79 \( 1 + 0.409T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−8.672040726079254107717999354862, −8.166236732672892918534618852460, −7.21718748685566793565173447621, −6.31064611626787532151448503121, −5.33078058536277384000252490006, −4.48839934083428248930867094401, −4.16727604521443044754487572500, −2.60301233676385610031142503513, −1.56666171496230014263661095266, 0, 1.56666171496230014263661095266, 2.60301233676385610031142503513, 4.16727604521443044754487572500, 4.48839934083428248930867094401, 5.33078058536277384000252490006, 6.31064611626787532151448503121, 7.21718748685566793565173447621, 8.166236732672892918534618852460, 8.672040726079254107717999354862

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