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Properties

Label 2-369600-1.1-c1-0-150
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 − 27-s + 2·29-s − 4·31-s + 33-s + 6·37-s + 2·39-s + 6·41-s + 12·43-s − 4·47-s + 49-s − 2·51-s + 6·53-s + 4·59-s + 10·61-s + 63-s − 4·67-s + 8·71-s + 2·73-s − 77-s + 81-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 − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 1.82·43-s − 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.125·63-s − 0.488·67-s + 0.949·71-s + 0.234·73-s − 0.113·77-s + 1/9·81-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.264778107\)
\(L(\frac12)\) \(\approx\) \(2.264778107\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.52189768526970, −12.11317587966287, −11.44255169355698, −11.21236705245872, −10.84398125279609, −10.14180635186455, −9.926922095043288, −9.433404169479260, −8.837473585473431, −8.412240849708998, −7.784836603337276, −7.441913868793655, −7.085323061828233, −6.419045185223060, −5.906513935902520, −5.512680076922696, −5.081891200804028, −4.506709510448034, −4.066622291245938, −3.552655577792858, −2.654304863138356, −2.453971611393991, −1.653519872645771, −0.9679756977627005, −0.4752768777385935, 0.4752768777385935, 0.9679756977627005, 1.653519872645771, 2.453971611393991, 2.654304863138356, 3.552655577792858, 4.066622291245938, 4.506709510448034, 5.081891200804028, 5.512680076922696, 5.906513935902520, 6.419045185223060, 7.085323061828233, 7.441913868793655, 7.784836603337276, 8.412240849708998, 8.837473585473431, 9.433404169479260, 9.926922095043288, 10.14180635186455, 10.84398125279609, 11.21236705245872, 11.44255169355698, 12.11317587966287, 12.52189768526970

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