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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(0.1097172691\)
\(L(\frac12)\) \(\approx\) \(0.1097172691\)
\(L(1)\) 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

−9.081254206913833905609832352449, −8.329084354636065712361325121479, −7.60186188996864652906585564427, −7.23270989218465963473258538260, −6.38485427210212267822223132203, −5.67964004878423604815519182042, −4.53666861509298569748209584891, −3.37947586417000630254820142603, −2.72447335993537685207080389095, −1.39655823689576388873916508275, 0.07362187985481995300420422382, 2.53042319459580765663924056754, 3.48762495864799037793911920137, 3.99604739238094352411035670149, 4.80315530352820844578770386146, 5.68733297317350510191030538498, 6.51329234329064185624640400641, 7.70415661237614244060168109222, 8.115260254411860894282057824453, 9.112897033414126146760211438537

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