Properties

Label 2-1150-1.1-c3-0-87
Degree $2$
Conductor $1150$
Sign $-1$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 8.27·3-s + 4·4-s − 16.5·6-s − 22.8·7-s − 8·8-s + 41.4·9-s − 0.987·11-s + 33.0·12-s + 4.22·13-s + 45.7·14-s + 16·16-s − 73.8·17-s − 82.8·18-s + 71.1·19-s − 189.·21-s + 1.97·22-s − 23·23-s − 66.1·24-s − 8.44·26-s + 119.·27-s − 91.4·28-s + 27.8·29-s − 136.·31-s − 32·32-s − 8.17·33-s + 147.·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.59·3-s + 0.5·4-s − 1.12·6-s − 1.23·7-s − 0.353·8-s + 1.53·9-s − 0.0270·11-s + 0.795·12-s + 0.0900·13-s + 0.872·14-s + 0.250·16-s − 1.05·17-s − 1.08·18-s + 0.859·19-s − 1.96·21-s + 0.0191·22-s − 0.208·23-s − 0.562·24-s − 0.0637·26-s + 0.849·27-s − 0.617·28-s + 0.178·29-s − 0.792·31-s − 0.176·32-s − 0.0431·33-s + 0.745·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
5 \( 1 \)
23 \( 1 + 23T \)
good3 \( 1 - 8.27T + 27T^{2} \)
7 \( 1 + 22.8T + 343T^{2} \)
11 \( 1 + 0.987T + 1.33e3T^{2} \)
13 \( 1 - 4.22T + 2.19e3T^{2} \)
17 \( 1 + 73.8T + 4.91e3T^{2} \)
19 \( 1 - 71.1T + 6.85e3T^{2} \)
29 \( 1 - 27.8T + 2.43e4T^{2} \)
31 \( 1 + 136.T + 2.97e4T^{2} \)
37 \( 1 - 201.T + 5.06e4T^{2} \)
41 \( 1 + 204.T + 6.89e4T^{2} \)
43 \( 1 - 54.1T + 7.95e4T^{2} \)
47 \( 1 - 41.4T + 1.03e5T^{2} \)
53 \( 1 + 428.T + 1.48e5T^{2} \)
59 \( 1 + 164.T + 2.05e5T^{2} \)
61 \( 1 - 188.T + 2.26e5T^{2} \)
67 \( 1 + 932.T + 3.00e5T^{2} \)
71 \( 1 + 263.T + 3.57e5T^{2} \)
73 \( 1 + 900.T + 3.89e5T^{2} \)
79 \( 1 + 956.T + 4.93e5T^{2} \)
83 \( 1 + 194.T + 5.71e5T^{2} \)
89 \( 1 + 213.T + 7.04e5T^{2} \)
97 \( 1 + 628.T + 9.12e5T^{2} \)
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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

−9.080040621836121955592649420877, −8.382079484891875862638448854984, −7.50700309127899877635142396951, −6.86526711296926276397211727307, −5.89394038801447478352882774056, −4.31029334352597233549777706537, −3.27762426248709137432515635730, −2.70944987903838765465218237153, −1.58604270428397208050931325882, 0, 1.58604270428397208050931325882, 2.70944987903838765465218237153, 3.27762426248709137432515635730, 4.31029334352597233549777706537, 5.89394038801447478352882774056, 6.86526711296926276397211727307, 7.50700309127899877635142396951, 8.382079484891875862638448854984, 9.080040621836121955592649420877

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