Properties

Label 2-6018-1.1-c1-0-133
Degree $2$
Conductor $6018$
Sign $-1$
Analytic cond. $48.0539$
Root an. cond. $6.93209$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 2·5-s − 6-s − 2·7-s + 8-s + 9-s + 2·10-s − 12-s − 2·14-s − 2·15-s + 16-s − 17-s + 18-s − 4·19-s + 2·20-s + 2·21-s + 2·23-s − 24-s − 25-s − 27-s − 2·28-s − 6·29-s − 2·30-s + 32-s − 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.436·21-s + 0.417·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.365·30-s + 0.176·32-s − 0.171·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6018\)    =    \(2 \cdot 3 \cdot 17 \cdot 59\)
Sign: $-1$
Analytic conductor: \(48.0539\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6018,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
17 \( 1 + T \)
59 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.44555785495948184673915622945, −6.67865737404974188529751328433, −6.27417612896666099481198760885, −5.59885687942236483338559735109, −4.98795530768093421309613031741, −4.09534991656052372361074351314, −3.31797213553165444268176328041, −2.33342941932956608657720016323, −1.54714495151500298486811239331, 0, 1.54714495151500298486811239331, 2.33342941932956608657720016323, 3.31797213553165444268176328041, 4.09534991656052372361074351314, 4.98795530768093421309613031741, 5.59885687942236483338559735109, 6.27417612896666099481198760885, 6.67865737404974188529751328433, 7.44555785495948184673915622945

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