Properties

Label 2-229320-1.1-c1-0-117
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 − 13-s + 2·17-s + 25-s + 2·29-s + 4·31-s + 6·37-s − 6·41-s + 4·43-s − 4·47-s + 10·53-s + 2·61-s − 65-s + 8·67-s − 4·71-s + 6·73-s − 8·79-s + 8·83-s + 2·85-s − 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.277·13-s + 0.485·17-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.583·47-s + 1.37·53-s + 0.256·61-s − 0.124·65-s + 0.977·67-s − 0.474·71-s + 0.702·73-s − 0.900·79-s + 0.878·83-s + 0.216·85-s − 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
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

−13.11765221303536, −12.79279992482162, −12.18142349576765, −11.81687379408235, −11.36769456776898, −10.80511043636849, −10.23833381329523, −9.983309021307615, −9.495965907212857, −8.966975743832673, −8.467928406526003, −7.987602902129988, −7.526768298916818, −6.904200852489066, −6.519387289047037, −5.951347710813214, −5.489857581739772, −4.937257090286985, −4.500415072706527, −3.797263825585870, −3.333319111265613, −2.507458045243975, −2.340515768216701, −1.360409075124873, −0.9306700426702671, 0, 0.9306700426702671, 1.360409075124873, 2.340515768216701, 2.507458045243975, 3.333319111265613, 3.797263825585870, 4.500415072706527, 4.937257090286985, 5.489857581739772, 5.951347710813214, 6.519387289047037, 6.904200852489066, 7.526768298916818, 7.987602902129988, 8.467928406526003, 8.966975743832673, 9.495965907212857, 9.983309021307615, 10.23833381329523, 10.80511043636849, 11.36769456776898, 11.81687379408235, 12.18142349576765, 12.79279992482162, 13.11765221303536

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