Properties

Label 2-189618-1.1-c1-0-35
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 + 4·7-s − 8-s + 9-s − 2·10-s + 11-s + 12-s − 4·14-s + 2·15-s + 16-s − 17-s − 18-s + 8·19-s + 2·20-s + 4·21-s − 22-s − 24-s − 25-s + 27-s + 4·28-s − 2·30-s − 10·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s − 1.06·14-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 + 0.872·21-s − 0.213·22-s − 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.755·28-s − 0.365·30-s − 1.79·31-s − 0.176·32-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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.45370367491817, −12.98507043348686, −12.28373956156174, −11.83496245913309, −11.37602911541277, −10.94702141483447, −10.54281126683757, −9.775797808120796, −9.600478500470079, −9.105900843001274, −8.657969232019008, −8.082082668031731, −7.702990777686812, −7.294597891386028, −6.744088165458649, −6.103314422157582, −5.432375540332678, −5.167012554606968, −4.593872247538289, −3.726710242671625, −3.321756269253147, −2.548218330219683, −1.929108780354849, −1.538029641429770, −1.166617014764185, 0, 1.166617014764185, 1.538029641429770, 1.929108780354849, 2.548218330219683, 3.321756269253147, 3.726710242671625, 4.593872247538289, 5.167012554606968, 5.432375540332678, 6.103314422157582, 6.744088165458649, 7.294597891386028, 7.702990777686812, 8.082082668031731, 8.657969232019008, 9.105900843001274, 9.600478500470079, 9.775797808120796, 10.54281126683757, 10.94702141483447, 11.37602911541277, 11.83496245913309, 12.28373956156174, 12.98507043348686, 13.45370367491817

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