Properties

Label 2-2352-1.1-c1-0-21
Degree $2$
Conductor $2352$
Sign $-1$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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 − 4·5-s + 9-s − 2·11-s + 6·13-s + 4·15-s + 4·17-s − 4·19-s − 2·23-s + 11·25-s − 27-s − 2·29-s + 2·33-s + 2·37-s − 6·39-s + 4·43-s − 4·45-s + 12·47-s − 4·51-s − 6·53-s + 8·55-s + 4·57-s − 8·59-s − 6·61-s − 24·65-s + 8·67-s + 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.417·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s + 0.348·33-s + 0.328·37-s − 0.960·39-s + 0.609·43-s − 0.596·45-s + 1.75·47-s − 0.560·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.04·59-s − 0.768·61-s − 2.97·65-s + 0.977·67-s + 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -1)\)

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

−8.360757557175675119343653904013, −7.88814631876587087290947240161, −7.18849610834707084119850003171, −6.22945629758055500711988795327, −5.49968891985445416500684424378, −4.33606078961091963255161029848, −3.90339449069204241119315605664, −2.95467819019400434754351026528, −1.22315778793251073811250041857, 0, 1.22315778793251073811250041857, 2.95467819019400434754351026528, 3.90339449069204241119315605664, 4.33606078961091963255161029848, 5.49968891985445416500684424378, 6.22945629758055500711988795327, 7.18849610834707084119850003171, 7.88814631876587087290947240161, 8.360757557175675119343653904013

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