Properties

Label 2-29400-1.1-c1-0-65
Degree $2$
Conductor $29400$
Sign $-1$
Analytic cond. $234.760$
Root an. cond. $15.3218$
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 − 2·13-s − 2·17-s − 6·19-s + 6·23-s − 27-s − 2·29-s − 2·31-s − 2·37-s + 2·39-s + 2·41-s + 8·43-s + 2·51-s + 6·57-s + 8·67-s − 6·69-s + 10·73-s − 8·79-s + 81-s − 12·83-s + 2·87-s − 6·89-s + 2·93-s + 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.554·13-s − 0.485·17-s − 1.37·19-s + 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.328·37-s + 0.320·39-s + 0.312·41-s + 1.21·43-s + 0.280·51-s + 0.794·57-s + 0.977·67-s − 0.722·69-s + 1.17·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s + 0.214·87-s − 0.635·89-s + 0.207·93-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(234.760\)
Root analytic conductor: \(15.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−15.47631132282305, −14.96116482293441, −14.39890308923198, −13.90171743770298, −13.06017328851610, −12.72782197247316, −12.44299583836325, −11.46849729428798, −11.25344459776178, −10.63328914628511, −10.18046068682561, −9.447381691533054, −8.945624656224861, −8.413366637617731, −7.617242108772611, −7.077403770491768, −6.601972609916160, −5.927642425238856, −5.351722537195738, −4.662220769984015, −4.212211854607302, −3.401967460514762, −2.517242461164176, −1.923933876116534, −0.8987335885612974, 0, 0.8987335885612974, 1.923933876116534, 2.517242461164176, 3.401967460514762, 4.212211854607302, 4.662220769984015, 5.351722537195738, 5.927642425238856, 6.601972609916160, 7.077403770491768, 7.617242108772611, 8.413366637617731, 8.945624656224861, 9.447381691533054, 10.18046068682561, 10.63328914628511, 11.25344459776178, 11.46849729428798, 12.44299583836325, 12.72782197247316, 13.06017328851610, 13.90171743770298, 14.39890308923198, 14.96116482293441, 15.47631132282305

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