Properties

Label 2-189618-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 14-s + 15-s + 16-s + 17-s − 18-s − 6·19-s + 20-s − 21-s + 22-s + 23-s − 24-s − 4·25-s + 27-s − 28-s + 29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s + 0.185·29-s − 0.182·30-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: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613876283\)
\(L(\frac12)\) \(\approx\) \(1.613876283\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.11722363098900, −12.74589412900033, −12.17546221528105, −11.69136868790435, −11.07647271338503, −10.59502469878524, −10.22326430022792, −9.771262654543181, −9.260764526551333, −8.966773165764449, −8.299546979378988, −7.969723256324882, −7.518427308341948, −6.846450001035903, −6.370070109770100, −6.055531571087546, −5.343857266090986, −4.717265387946857, −4.135953169293809, −3.456696571043017, −3.002927719565409, −2.201841870776010, −2.039735605366874, −1.184647462245264, −0.3961117856238029, 0.3961117856238029, 1.184647462245264, 2.039735605366874, 2.201841870776010, 3.002927719565409, 3.456696571043017, 4.135953169293809, 4.717265387946857, 5.343857266090986, 6.055531571087546, 6.370070109770100, 6.846450001035903, 7.518427308341948, 7.969723256324882, 8.299546979378988, 8.966773165764449, 9.260764526551333, 9.771262654543181, 10.22326430022792, 10.59502469878524, 11.07647271338503, 11.69136868790435, 12.17546221528105, 12.74589412900033, 13.11722363098900

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