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Properties

Label 2-2448-1.1-c1-0-37
Degree $2$
Conductor $2448$
Sign $-1$
Analytic cond. $19.5473$
Root an. cond. $4.42124$
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  + 2·5-s − 4·11-s − 2·13-s − 17-s − 4·19-s − 25-s + 10·29-s − 8·31-s − 2·37-s − 10·41-s − 12·43-s − 7·49-s − 6·53-s − 8·55-s + 12·59-s − 10·61-s − 4·65-s + 12·67-s + 10·73-s + 8·79-s + 4·83-s − 2·85-s + 6·89-s − 8·95-s − 14·97-s + 10·101-s + 8·103-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.20·11-s − 0.554·13-s − 0.242·17-s − 0.917·19-s − 1/5·25-s + 1.85·29-s − 1.43·31-s − 0.328·37-s − 1.56·41-s − 1.82·43-s − 49-s − 0.824·53-s − 1.07·55-s + 1.56·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s + 1.17·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.820·95-s − 1.42·97-s + 0.995·101-s + 0.788·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2448\)    =    \(2^{4} \cdot 3^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(19.5473\)
Root analytic conductor: \(4.42124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2448,\ (\ :1/2),\ -1)\)

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 \( 1 \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.467961047217185243736660174457, −7.955859097314133791044966496465, −6.84651712139757264196140765804, −6.33169466753969442189844266176, −5.21730475246735557714050254853, −4.90718026477218821976639724467, −3.55640376879290048858690413154, −2.51899204061915291020295701141, −1.78111713369696792468537479439, 0, 1.78111713369696792468537479439, 2.51899204061915291020295701141, 3.55640376879290048858690413154, 4.90718026477218821976639724467, 5.21730475246735557714050254853, 6.33169466753969442189844266176, 6.84651712139757264196140765804, 7.955859097314133791044966496465, 8.467961047217185243736660174457

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