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Properties

Label 2-1872-117.38-c0-0-0
Degree $2$
Conductor $1872$
Sign $0.342 + 0.939i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)13-s − 1.73i·17-s + (1.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)39-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (−1.49 + 0.866i)51-s − 1.73i·53-s + (−0.5 + 0.866i)61-s + (−1.5 − 0.866i)69-s − 0.999·75-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)13-s − 1.73i·17-s + (1.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)39-s + (−0.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (−1.49 + 0.866i)51-s − 1.73i·53-s + (−0.5 + 0.866i)61-s + (−1.5 − 0.866i)69-s − 0.999·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9591815064\)
\(L(\frac12)\) \(\approx\) \(0.9591815064\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.73iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.097780563713567726549849736306, −8.505145904525347702543465498217, −7.53122334854734695431285372606, −6.78770351323561841005838844190, −6.37034516380259925327355654741, −5.14692531941424087997390354900, −4.64321316569666442195385008676, −3.14810553431469102464249752767, −2.20444652337649106729145775900, −0.879228175509111654307510777508, 1.32926659628246428481655672995, 3.08502261876200165825194095500, 3.71169189504131754513694289593, 4.74435623128006893595535261140, 5.56215959007961297242509410231, 6.15764457225174523344619586997, 7.15519439432741010164565192983, 8.166747096623022822404809626126, 8.906906158652979269365749605768, 9.561395443769373387670939557851

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