Properties

Label 2-189618-1.1-c1-0-32
Degree $2$
Conductor $189618$
Sign $-1$
Analytic cond. $1514.10$
Root an. cond. $38.9115$
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 + 7-s + 8-s + 9-s + 2·10-s − 11-s − 12-s + 14-s − 2·15-s + 16-s + 17-s + 18-s − 8·19-s + 2·20-s − 21-s − 22-s + 2·23-s − 24-s − 25-s − 27-s + 28-s + 2·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.218·21-s − 0.213·22-s + 0.417·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189618\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1514.10\)
Root analytic conductor: \(38.9115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 5 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.16110789291869, −13.03091088135471, −12.48469293987971, −11.95369433096562, −11.58851179512394, −10.90063346320320, −10.66240971253296, −10.20089735219130, −9.730108465764792, −9.108759469251948, −8.604969070678366, −7.929471293781588, −7.626422939525499, −6.833046413711613, −6.364253256675021, −6.044805434606021, −5.666087378393754, −4.902712778013559, −4.587431644397434, −4.189770951249519, −3.356692073250828, −2.665808714043236, −2.239839674168080, −1.599506868134045, −0.9761510297914423, 0, 0.9761510297914423, 1.599506868134045, 2.239839674168080, 2.665808714043236, 3.356692073250828, 4.189770951249519, 4.587431644397434, 4.902712778013559, 5.666087378393754, 6.044805434606021, 6.364253256675021, 6.833046413711613, 7.626422939525499, 7.929471293781588, 8.604969070678366, 9.108759469251948, 9.730108465764792, 10.20089735219130, 10.66240971253296, 10.90063346320320, 11.58851179512394, 11.95369433096562, 12.48469293987971, 13.03091088135471, 13.16110789291869

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