Properties

Label 2-369600-1.1-c1-0-107
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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  − 3-s − 7-s + 9-s + 11-s + 2·13-s − 2·17-s + 21-s + 6·23-s − 27-s + 2·29-s − 6·31-s − 33-s + 2·37-s − 2·39-s − 8·43-s + 4·47-s + 49-s + 2·51-s + 6·53-s − 6·59-s − 8·61-s − 63-s + 10·67-s − 6·69-s − 6·73-s − 77-s + 16·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.174·33-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.781·59-s − 1.02·61-s − 0.125·63-s + 1.22·67-s − 0.722·69-s − 0.702·73-s − 0.113·77-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784564026\)
\(L(\frac12)\) \(\approx\) \(1.784564026\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.58425380350694, −12.00357151407068, −11.51691720203866, −11.20479301312921, −10.71458265723392, −10.37734590522338, −9.772835365410875, −9.364395555329247, −8.740598988588879, −8.675071955916477, −7.828383339152418, −7.341467257260915, −6.917943180441991, −6.477297154906892, −6.027158657430160, −5.571888230561487, −4.946024343126632, −4.619379081930823, −3.933270142615095, −3.478394480581786, −2.976190736731861, −2.256873782212980, −1.643630231519791, −1.013775438608719, −0.4081525300049825, 0.4081525300049825, 1.013775438608719, 1.643630231519791, 2.256873782212980, 2.976190736731861, 3.478394480581786, 3.933270142615095, 4.619379081930823, 4.946024343126632, 5.571888230561487, 6.027158657430160, 6.477297154906892, 6.917943180441991, 7.341467257260915, 7.828383339152418, 8.675071955916477, 8.740598988588879, 9.364395555329247, 9.772835365410875, 10.37734590522338, 10.71458265723392, 11.20479301312921, 11.51691720203866, 12.00357151407068, 12.58425380350694

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