Properties

Label 2-160446-1.1-c1-0-47
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 + 4·5-s + 6-s + 8-s + 9-s + 4·10-s + 12-s + 13-s + 4·15-s + 16-s − 17-s + 18-s − 2·19-s + 4·20-s − 4·23-s + 24-s + 11·25-s + 26-s + 27-s − 2·29-s + 4·30-s + 32-s − 34-s + 36-s − 6·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s + 0.277·13-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.894·20-s − 0.834·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s − 0.371·29-s + 0.730·30-s + 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.986·37-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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.53296810025496, −13.01728025719091, −12.90031628351728, −12.33039711500195, −11.59234594748441, −11.19642444917080, −10.50577524016775, −10.17252941999898, −9.736016324688556, −9.333508649529292, −8.613576263720946, −8.386394955111233, −7.671077964875244, −6.922321960565114, −6.570250586593932, −6.187975439608015, −5.556291850624540, −5.156836031895313, −4.648089913449150, −3.891835445270439, −3.413839746778142, −2.713604897297680, −2.258227560422209, −1.669624547922325, −1.345772336058013, 0, 1.345772336058013, 1.669624547922325, 2.258227560422209, 2.713604897297680, 3.413839746778142, 3.891835445270439, 4.648089913449150, 5.156836031895313, 5.556291850624540, 6.187975439608015, 6.570250586593932, 6.922321960565114, 7.671077964875244, 8.386394955111233, 8.613576263720946, 9.333508649529292, 9.736016324688556, 10.17252941999898, 10.50577524016775, 11.19642444917080, 11.59234594748441, 12.33039711500195, 12.90031628351728, 13.01728025719091, 13.53296810025496

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