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Properties

Label 2-1175-1.1-c1-0-36
Degree $2$
Conductor $1175$
Sign $-1$
Analytic cond. $9.38242$
Root an. cond. $3.06307$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s + 4·6-s + 2·7-s + 9-s − 4·12-s − 3·13-s − 4·14-s − 4·16-s − 2·18-s − 4·19-s − 4·21-s − 23-s + 6·26-s + 4·27-s + 4·28-s + 8·29-s + 6·31-s + 8·32-s + 2·36-s + 6·37-s + 8·38-s + 6·39-s − 2·41-s + 8·42-s − 9·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s + 0.755·7-s + 1/3·9-s − 1.15·12-s − 0.832·13-s − 1.06·14-s − 16-s − 0.471·18-s − 0.917·19-s − 0.872·21-s − 0.208·23-s + 1.17·26-s + 0.769·27-s + 0.755·28-s + 1.48·29-s + 1.07·31-s + 1.41·32-s + 1/3·36-s + 0.986·37-s + 1.29·38-s + 0.960·39-s − 0.312·41-s + 1.23·42-s − 1.37·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1175\)    =    \(5^{2} \cdot 47\)
Sign: $-1$
Analytic conductor: \(9.38242\)
Root analytic conductor: \(3.06307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1175,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
47 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−9.502879417426197200546849682577, −8.356322250980029058390853722909, −8.067233007582536882648676645536, −6.88075125273545451287432246129, −6.31345995783390560959863226517, −5.04421225255399783722139869844, −4.49670803018523313192416115565, −2.54092614197265454021832247774, −1.26211787338682255500293713315, 0, 1.26211787338682255500293713315, 2.54092614197265454021832247774, 4.49670803018523313192416115565, 5.04421225255399783722139869844, 6.31345995783390560959863226517, 6.88075125273545451287432246129, 8.067233007582536882648676645536, 8.356322250980029058390853722909, 9.502879417426197200546849682577

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