Properties

Label 2-1104-276.275-c0-0-1
Degree $2$
Conductor $1104$
Sign $1$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.41·5-s + 1.41·7-s + 9-s − 1.41·15-s − 1.41·17-s − 1.41·19-s − 1.41·21-s + 23-s + 1.00·25-s − 27-s + 2.00·35-s + 1.41·43-s + 1.41·45-s + 1.00·49-s + 1.41·51-s − 1.41·53-s + 1.41·57-s + 1.41·63-s − 1.41·67-s − 69-s + 2·71-s − 2·73-s − 1.00·75-s − 1.41·79-s + 81-s − 2.00·85-s + ⋯
L(s)  = 1  − 3-s + 1.41·5-s + 1.41·7-s + 9-s − 1.41·15-s − 1.41·17-s − 1.41·19-s − 1.41·21-s + 23-s + 1.00·25-s − 27-s + 2.00·35-s + 1.41·43-s + 1.41·45-s + 1.00·49-s + 1.41·51-s − 1.41·53-s + 1.41·57-s + 1.41·63-s − 1.41·67-s − 69-s + 2·71-s − 2·73-s − 1.00·75-s − 1.41·79-s + 81-s − 2.00·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (1103, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087930335\)
\(L(\frac12)\) \(\approx\) \(1.087930335\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.25929506811972435766919237761, −9.256367422211388818371432824175, −8.569445135946363737269325911629, −7.38667988818145706654476376800, −6.47367438641744663291352348523, −5.84464475579063548732857058274, −4.89998424796880393431847506936, −4.38145930497602692576777313894, −2.32162399434700337058209002122, −1.50727428499267185578308857284, 1.50727428499267185578308857284, 2.32162399434700337058209002122, 4.38145930497602692576777313894, 4.89998424796880393431847506936, 5.84464475579063548732857058274, 6.47367438641744663291352348523, 7.38667988818145706654476376800, 8.569445135946363737269325911629, 9.256367422211388818371432824175, 10.25929506811972435766919237761

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