Properties

Label 2-160446-1.1-c1-0-10
Degree $2$
Conductor $160446$
Sign $-1$
Analytic cond. $1281.16$
Root an. cond. $35.7934$
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 − 8-s + 9-s + 2·10-s − 12-s − 13-s + 2·15-s + 16-s + 17-s − 18-s − 8·19-s − 2·20-s − 4·23-s + 24-s − 25-s + 26-s − 27-s − 6·29-s − 2·30-s − 4·31-s − 32-s − 34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.277·13-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 − 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.11·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160446\)    =    \(2 \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(1281.16\)
Root analytic conductor: \(35.7934\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{160446} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 160446,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.34942083765397, −12.81856277525104, −12.46516436422483, −12.02619970074055, −11.53351584790773, −11.04336042386133, −10.80911649260997, −10.17846086432627, −9.742428685148771, −9.252896416293510, −8.581348356154453, −8.132393717121644, −7.886797156049396, −7.116237362333591, −6.824288665467742, −6.312594480656458, −5.579299843722933, −5.282554992864097, −4.425508833791894, −3.880855435402582, −3.651114057509720, −2.644898860642229, −2.019454496338000, −1.533337118372144, −0.4701811241477430, 0, 0.4701811241477430, 1.533337118372144, 2.019454496338000, 2.644898860642229, 3.651114057509720, 3.880855435402582, 4.425508833791894, 5.282554992864097, 5.579299843722933, 6.312594480656458, 6.824288665467742, 7.116237362333591, 7.886797156049396, 8.132393717121644, 8.581348356154453, 9.252896416293510, 9.742428685148771, 10.17846086432627, 10.80911649260997, 11.04336042386133, 11.53351584790773, 12.02619970074055, 12.46516436422483, 12.81856277525104, 13.34942083765397

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