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Properties

Label 2-2432-19.18-c0-0-4
Degree $2$
Conductor $2432$
Sign $i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·3-s − 5-s + 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s − 1.41i·33-s − 35-s + 1.41i·37-s − 1.41i·41-s + ⋯
L(s)  = 1  − 1.41i·3-s − 5-s + 7-s − 1.00·9-s + 11-s + 1.41i·15-s + 17-s + 19-s − 1.41i·21-s − 1.41i·29-s + 1.41i·31-s − 1.41i·33-s − 35-s + 1.41i·37-s − 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2432,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227024270\)
\(L(\frac12)\) \(\approx\) \(1.227024270\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−8.491825895088522119862322109607, −8.101600362273263136931656871480, −7.41026071020686152316376256933, −6.92236239437042172429745359214, −5.97652372593103427789680967519, −5.04787558198727184179532020758, −4.05007910976000626973555023779, −3.13396020373967825159368495513, −1.79225205749975178118609305358, −1.02223138353239144806784506481, 1.39103162093925498407498392169, 3.12035050977351861637054268404, 3.81956715260758046083100426242, 4.44063626358879832058191476387, 5.14919372078583084671490445168, 5.98367756253079699687046587307, 7.34030527649772624873804638411, 7.78381254635765515907635843900, 8.705667051627909153195451006235, 9.344948254784760498386269547587

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