Properties

Label 2-160446-1.1-c1-0-17
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 − 5-s − 6-s − 2·7-s − 8-s + 9-s + 10-s + 12-s − 13-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 4·19-s − 20-s − 2·21-s − 9·23-s − 24-s − 4·25-s + 26-s + 27-s − 2·28-s − 6·29-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 − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.436·21-s − 1.87·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + 0.182·30-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: Trivial
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 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.27610350204723, −12.99818159463945, −12.73306189314476, −11.99085001661631, −11.46416121876549, −11.28570648279372, −10.45245850973054, −10.00320060730405, −9.741610919299506, −9.232149741804878, −8.662155188826946, −8.230353856176489, −7.787021458974921, −7.344795518007869, −6.773188977751695, −6.318696176682059, −5.736242165218714, −5.215661947272290, −4.184523159874898, −3.914645029080987, −3.495462877231697, −2.640583993364472, −2.098822791177851, −1.753459987456170, −0.5763572115865194, 0, 0.5763572115865194, 1.753459987456170, 2.098822791177851, 2.640583993364472, 3.495462877231697, 3.914645029080987, 4.184523159874898, 5.215661947272290, 5.736242165218714, 6.318696176682059, 6.773188977751695, 7.344795518007869, 7.787021458974921, 8.230353856176489, 8.662155188826946, 9.232149741804878, 9.741610919299506, 10.00320060730405, 10.45245850973054, 11.28570648279372, 11.46416121876549, 11.99085001661631, 12.73306189314476, 12.99818159463945, 13.27610350204723

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