Properties

Label 2-348480-1.1-c1-0-84
Degree $2$
Conductor $348480$
Sign $1$
Analytic cond. $2782.62$
Root an. cond. $52.7506$
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 + 2·7-s + 2·13-s − 2·19-s + 25-s − 8·31-s − 2·35-s − 2·37-s − 2·43-s − 3·49-s + 6·53-s + 12·59-s + 2·61-s − 2·65-s − 4·67-s − 2·73-s − 10·79-s − 12·83-s + 6·89-s + 4·91-s + 2·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 0.554·13-s − 0.458·19-s + 1/5·25-s − 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.304·43-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.248·65-s − 0.488·67-s − 0.234·73-s − 1.12·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s + 0.205·95-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348480\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(2782.62\)
Root analytic conductor: \(52.7506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{348480} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 348480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.860832098\)
\(L(\frac12)\) \(\approx\) \(1.860832098\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−12.58070086066876, −11.97841391353438, −11.60014328175676, −11.25846187678970, −10.80456686295464, −10.35019917552988, −9.949978044591141, −9.187004837247783, −8.892087872187767, −8.390940070509748, −8.037273591973061, −7.513253923940526, −7.013244216913822, −6.642281765557586, −5.957405740511835, −5.428386504848949, −5.141954570141147, −4.309711040347916, −4.139614307457958, −3.476848389824674, −2.987299991794290, −2.205379828976531, −1.746168947412926, −1.140803446789325, −0.3703842743846824, 0.3703842743846824, 1.140803446789325, 1.746168947412926, 2.205379828976531, 2.987299991794290, 3.476848389824674, 4.139614307457958, 4.309711040347916, 5.141954570141147, 5.428386504848949, 5.957405740511835, 6.642281765557586, 7.013244216913822, 7.513253923940526, 8.037273591973061, 8.390940070509748, 8.892087872187767, 9.187004837247783, 9.949978044591141, 10.35019917552988, 10.80456686295464, 11.25846187678970, 11.60014328175676, 11.97841391353438, 12.58070086066876

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