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Properties

Label 2-1872-208.51-c0-0-2
Degree $2$
Conductor $1872$
Sign $0.923 + 0.382i$
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  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.541 + 0.541i)5-s + (0.382 − 0.923i)8-s + (0.707 + 0.292i)10-s + (−0.541 + 0.541i)11-s + (0.707 + 0.707i)13-s − i·16-s + 0.765·20-s + (−0.292 + 0.707i)22-s − 0.414i·25-s + (0.923 + 0.382i)26-s + (−0.382 − 0.923i)32-s + (0.707 − 0.292i)40-s − 0.765·41-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.541 + 0.541i)5-s + (0.382 − 0.923i)8-s + (0.707 + 0.292i)10-s + (−0.541 + 0.541i)11-s + (0.707 + 0.707i)13-s − i·16-s + 0.765·20-s + (−0.292 + 0.707i)22-s − 0.414i·25-s + (0.923 + 0.382i)26-s + (−0.382 − 0.923i)32-s + (0.707 − 0.292i)40-s − 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.177816748\)
\(L(\frac12)\) \(\approx\) \(2.177816748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
3 \( 1 \)
13 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + 0.765T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 0.765iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
89 \( 1 + 1.84T + T^{2} \)
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

−9.636087941770067442913858607211, −8.687697273127681205114119419709, −7.54614478949998433734026428295, −6.73679523140277139872964503385, −6.16070767542309768593672075791, −5.27235756799664577362089609121, −4.42957884129729841586529078190, −3.48747140725327233743232613661, −2.50439798157545175101046151325, −1.64845607135786557056170392997, 1.56416235192031470248687706679, 2.85991859565469638922578385891, 3.63596700927309134359909146750, 4.78987707332713478440354933360, 5.44376736328926327040473540444, 6.06899684750243352898975042395, 6.91438459423210596832551377373, 7.980166874251283940163855866120, 8.406070256309084620478632222397, 9.376114407207631243297890298767

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