Properties

Label 2-2151-1.1-c1-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.02·2-s + 2.10·4-s + 3.48·5-s + 1.37·7-s − 0.213·8-s − 7.05·10-s + 0.191·11-s + 4.24·13-s − 2.77·14-s − 3.77·16-s + 4.07·17-s − 5.16·19-s + 7.32·20-s − 0.388·22-s + 0.978·23-s + 7.11·25-s − 8.59·26-s + 2.88·28-s + 10.0·29-s − 4.04·31-s + 8.08·32-s − 8.26·34-s + 4.76·35-s + 2.15·37-s + 10.4·38-s − 0.744·40-s + 0.0317·41-s + ⋯
L(s)  = 1  − 1.43·2-s + 1.05·4-s + 1.55·5-s + 0.517·7-s − 0.0755·8-s − 2.23·10-s + 0.0577·11-s + 1.17·13-s − 0.741·14-s − 0.944·16-s + 0.989·17-s − 1.18·19-s + 1.63·20-s − 0.0827·22-s + 0.204·23-s + 1.42·25-s − 1.68·26-s + 0.545·28-s + 1.85·29-s − 0.727·31-s + 1.42·32-s − 1.41·34-s + 0.806·35-s + 0.353·37-s + 1.69·38-s − 0.117·40-s + 0.00496·41-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: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.375853216\)
\(L(\frac12)\) \(\approx\) \(1.375853216\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.02T + 2T^{2} \)
5 \( 1 - 3.48T + 5T^{2} \)
7 \( 1 - 1.37T + 7T^{2} \)
11 \( 1 - 0.191T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 - 4.07T + 17T^{2} \)
19 \( 1 + 5.16T + 19T^{2} \)
23 \( 1 - 0.978T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + 4.04T + 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 - 0.0317T + 41T^{2} \)
43 \( 1 - 2.77T + 43T^{2} \)
47 \( 1 - 0.680T + 47T^{2} \)
53 \( 1 - 4.91T + 53T^{2} \)
59 \( 1 - 3.94T + 59T^{2} \)
61 \( 1 - 3.63T + 61T^{2} \)
67 \( 1 - 5.37T + 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 + 7.76T + 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 2.22T + 83T^{2} \)
89 \( 1 + 9.94T + 89T^{2} \)
97 \( 1 + 4.84T + 97T^{2} \)
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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

−8.912774455415244482367454501773, −8.622243647072224899491188177203, −7.81147189441310216630627097682, −6.77729511377009609566109783554, −6.15664039348435529420007979032, −5.33289842055830907454995179795, −4.24904153264549455566694763481, −2.74790510535910751762534630128, −1.75565091585114491665158062310, −1.06305757238960019216776506424, 1.06305757238960019216776506424, 1.75565091585114491665158062310, 2.74790510535910751762534630128, 4.24904153264549455566694763481, 5.33289842055830907454995179795, 6.15664039348435529420007979032, 6.77729511377009609566109783554, 7.81147189441310216630627097682, 8.622243647072224899491188177203, 8.912774455415244482367454501773

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