Properties

Label 2-229320-1.1-c1-0-112
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
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  + 5-s + 2·11-s + 13-s − 2·17-s − 6·19-s − 4·23-s + 25-s + 4·29-s + 4·31-s + 6·37-s + 10·41-s + 8·43-s + 2·53-s + 2·55-s − 8·59-s + 4·61-s + 65-s − 10·67-s − 12·71-s + 2·73-s − 6·83-s − 2·85-s − 14·89-s − 6·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.603·11-s + 0.277·13-s − 0.485·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s + 0.742·29-s + 0.718·31-s + 0.986·37-s + 1.56·41-s + 1.21·43-s + 0.274·53-s + 0.269·55-s − 1.04·59-s + 0.512·61-s + 0.124·65-s − 1.22·67-s − 1.42·71-s + 0.234·73-s − 0.658·83-s − 0.216·85-s − 1.48·89-s − 0.615·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.10400853071999, −12.73886410207034, −12.28752762945061, −11.79656287600655, −11.24540589130892, −10.85193569822900, −10.38464550436291, −9.924239171938102, −9.407664723255501, −8.912823477720213, −8.579596579171160, −8.015382216161574, −7.502291251274926, −6.902893348121285, −6.383176766314623, −5.935448143827582, −5.783768407717566, −4.730582280845178, −4.304653753182408, −4.136433430938942, −3.231198808954317, −2.579702828345891, −2.222100085696617, −1.455371228996576, −0.8799780915409866, 0, 0.8799780915409866, 1.455371228996576, 2.222100085696617, 2.579702828345891, 3.231198808954317, 4.136433430938942, 4.304653753182408, 4.730582280845178, 5.783768407717566, 5.935448143827582, 6.383176766314623, 6.902893348121285, 7.502291251274926, 8.015382216161574, 8.579596579171160, 8.912823477720213, 9.407664723255501, 9.924239171938102, 10.38464550436291, 10.85193569822900, 11.24540589130892, 11.79656287600655, 12.28752762945061, 12.73886410207034, 13.10400853071999

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