Properties

Label 2-48e2-1.1-c1-0-21
Degree $2$
Conductor $2304$
Sign $-1$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  − 4·5-s + 4·13-s + 2·17-s + 11·25-s − 4·29-s − 12·37-s + 10·41-s − 7·49-s − 4·53-s − 12·61-s − 16·65-s − 6·73-s − 8·85-s − 10·89-s − 18·97-s − 20·101-s + 20·109-s + 14·113-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.10·13-s + 0.485·17-s + 11/5·25-s − 0.742·29-s − 1.97·37-s + 1.56·41-s − 49-s − 0.549·53-s − 1.53·61-s − 1.98·65-s − 0.702·73-s − 0.867·85-s − 1.05·89-s − 1.82·97-s − 1.99·101-s + 1.91·109-s + 1.31·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.518580926433601982373809526294, −7.83885085572920869613116207995, −7.27777410584902448752390598516, −6.39063738730332703469978407006, −5.39010496228517083389897023205, −4.38564257449405141873091168941, −3.71637470346215322038560926740, −3.04380780885898707635692948137, −1.36924256495576692234243060544, 0, 1.36924256495576692234243060544, 3.04380780885898707635692948137, 3.71637470346215322038560926740, 4.38564257449405141873091168941, 5.39010496228517083389897023205, 6.39063738730332703469978407006, 7.27777410584902448752390598516, 7.83885085572920869613116207995, 8.518580926433601982373809526294

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