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Properties

Label 2-3864-3864.275-c0-0-0
Degree $2$
Conductor $3864$
Sign $-0.0633 - 0.997i$
Analytic cond. $1.92838$
Root an. cond. $1.38866$
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)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 + 0.866i)12-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)18-s + (−0.499 + 0.866i)21-s − 0.999·22-s + (−0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(1.92838\)
Root analytic conductor: \(1.38866\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3864,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4947594199\)
\(L(\frac12)\) \(\approx\) \(0.4947594199\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.635164471155224299609633146046, −7.81883514142034311086079743865, −7.40467964676419264560926731991, −6.68828042689835933005012048544, −6.17193879448464788014814790169, −5.41192175869626677242436999120, −4.47959914242293995394590980714, −3.63311580300619851180057504939, −1.98430280576089188041396917564, −1.16918345609786118996748633236, 0.39728433309302425016243066789, 1.98439924577398963463871647379, 3.14460965188376305172179032263, 3.56565743517970142788511140951, 4.53439156329074768479390790282, 5.46825177460587599965927425734, 6.01626234519409887287532324077, 7.05797140902649174918237462276, 8.006432103935474332582634242024, 8.936458112476385394368884422230

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