Properties

Label 2-110400-1.1-c1-0-157
Degree $2$
Conductor $110400$
Sign $-1$
Analytic cond. $881.548$
Root an. cond. $29.6908$
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  + 3-s + 9-s − 4·11-s − 2·13-s + 6·17-s − 4·19-s + 23-s + 27-s + 2·29-s − 4·33-s − 2·37-s − 2·39-s + 10·41-s − 4·43-s − 7·49-s + 6·51-s + 6·53-s − 4·57-s + 4·59-s + 10·61-s − 12·67-s + 69-s − 8·71-s − 10·73-s − 8·79-s + 81-s − 4·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.208·23-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.328·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s − 49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s + 1.28·61-s − 1.46·67-s + 0.120·69-s − 0.949·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s − 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(881.548\)
Root analytic conductor: \(29.6908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{110400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 110400,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.94815678217115, −13.25704922713391, −13.01571114707408, −12.57609046430002, −11.97378392694715, −11.58805198612104, −10.78190693999395, −10.37998241399401, −10.06481677419695, −9.518274842457173, −8.939142742686950, −8.303382796532214, −8.076441200191462, −7.331596296734647, −7.209679790669506, −6.315278803573169, −5.741545963134081, −5.279255564123928, −4.646856701895953, −4.162765084579965, −3.370023665728708, −2.896880302185470, −2.396779848354191, −1.709183204745719, −0.8841526813713899, 0, 0.8841526813713899, 1.709183204745719, 2.396779848354191, 2.896880302185470, 3.370023665728708, 4.162765084579965, 4.646856701895953, 5.279255564123928, 5.741545963134081, 6.315278803573169, 7.209679790669506, 7.331596296734647, 8.076441200191462, 8.303382796532214, 8.939142742686950, 9.518274842457173, 10.06481677419695, 10.37998241399401, 10.78190693999395, 11.58805198612104, 11.97378392694715, 12.57609046430002, 13.01571114707408, 13.25704922713391, 13.94815678217115

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