Properties

Label 2-5520-1.1-c1-0-73
Degree $2$
Conductor $5520$
Sign $-1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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  + 3-s + 5-s − 4·7-s + 9-s − 2·11-s + 4·13-s + 15-s − 6·17-s + 4·19-s − 4·21-s + 23-s + 25-s + 27-s + 8·29-s − 8·31-s − 2·33-s − 4·35-s − 10·37-s + 4·39-s + 6·41-s − 6·43-s + 45-s + 4·47-s + 9·49-s − 6·51-s − 14·53-s − 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s + 0.640·39-s + 0.937·41-s − 0.914·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.840·51-s − 1.92·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.85018715308465521126964444333, −6.87569384061775128550176186567, −6.54850821138916141178301057259, −5.74212435350864226418090293196, −4.89591711814810999541117357731, −3.85712037806955148081109502780, −3.19535878459479555961756788698, −2.55063514391836867296888573113, −1.44185294694951357160000267790, 0, 1.44185294694951357160000267790, 2.55063514391836867296888573113, 3.19535878459479555961756788698, 3.85712037806955148081109502780, 4.89591711814810999541117357731, 5.74212435350864226418090293196, 6.54850821138916141178301057259, 6.87569384061775128550176186567, 7.85018715308465521126964444333

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