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Properties

Label 2-2432-19.18-c0-0-7
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

\(L(\frac{1}{2})\) \(\approx\) \(1.510196287\)
\(L(\frac12)\) \(\approx\) \(1.510196287\)
\(L(1)\) 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

−8.653094923891008206358254448164, −7.87590664909769522954618263760, −7.70912239580716861992538208203, −6.53998668565952309128283896751, −5.94765366757711923672069878061, −5.27815568994135145061491950944, −4.23152317681079038390802354369, −2.59245854390906704341164083930, −2.10761664994268906131972631424, −1.10537673504947403609078840838, 1.68289640088222024848000251785, 2.73247974143644809381584538023, 3.79306020109889461278852983434, 4.67957744370422299727739078929, 5.39273852645886723190454549553, 5.73688091571638148801460200408, 7.09110441015822124429419541493, 7.938191712881340308333911530042, 8.872935448251216144919976606266, 9.250953202723730190323717888269

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