Properties

Label 2-212940-1.1-c1-0-3
Degree $2$
Conductor $212940$
Sign $1$
Analytic cond. $1700.33$
Root an. cond. $41.2351$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 5·11-s + 17-s − 6·19-s − 6·23-s + 25-s + 9·29-s + 4·31-s − 35-s − 2·37-s − 4·41-s + 10·43-s − 47-s + 49-s − 4·53-s + 5·55-s − 8·59-s − 8·61-s − 12·67-s + 8·71-s − 2·73-s − 5·77-s + 13·79-s − 4·83-s − 85-s + 4·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.50·11-s + 0.242·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.624·41-s + 1.52·43-s − 0.145·47-s + 1/7·49-s − 0.549·53-s + 0.674·55-s − 1.04·59-s − 1.02·61-s − 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.569·77-s + 1.46·79-s − 0.439·83-s − 0.108·85-s + 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(212940\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1700.33\)
Root analytic conductor: \(41.2351\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{212940} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 212940,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8616839449\)
\(L(\frac12)\) \(\approx\) \(0.8616839449\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
good11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.88513997140703, −12.50186207421059, −12.12879338642448, −11.69811928306624, −11.02295317677134, −10.58861882708040, −10.32789375126937, −9.937771182919254, −9.082555183062187, −8.710477364410921, −8.128877822166809, −7.819936880813398, −7.549149719314070, −6.688859030049348, −6.251572700765978, −5.842579493258570, −5.086253966752510, −4.648736869288060, −4.342318377198110, −3.585091312486331, −2.970795474261726, −2.435522852831319, −1.952039482322335, −1.096133519358092, −0.2746415559780871, 0.2746415559780871, 1.096133519358092, 1.952039482322335, 2.435522852831319, 2.970795474261726, 3.585091312486331, 4.342318377198110, 4.648736869288060, 5.086253966752510, 5.842579493258570, 6.251572700765978, 6.688859030049348, 7.549149719314070, 7.819936880813398, 8.128877822166809, 8.710477364410921, 9.082555183062187, 9.937771182919254, 10.32789375126937, 10.58861882708040, 11.02295317677134, 11.69811928306624, 12.12879338642448, 12.50186207421059, 12.88513997140703

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