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Properties

Label 2-235200-1.1-c1-0-208
Degree $2$
Conductor $235200$
Sign $-1$
Analytic cond. $1878.08$
Root an. cond. $43.3368$
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 − 6·11-s − 2·13-s − 4·17-s + 8·19-s − 8·23-s − 27-s − 8·29-s − 2·31-s + 6·33-s + 2·39-s − 6·41-s + 2·43-s − 2·47-s + 4·51-s + 6·53-s − 8·57-s − 12·59-s − 10·61-s − 2·67-s + 8·69-s − 12·71-s + 10·73-s − 8·79-s + 81-s + 16·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 0.970·17-s + 1.83·19-s − 1.66·23-s − 0.192·27-s − 1.48·29-s − 0.359·31-s + 1.04·33-s + 0.320·39-s − 0.937·41-s + 0.304·43-s − 0.291·47-s + 0.560·51-s + 0.824·53-s − 1.05·57-s − 1.56·59-s − 1.28·61-s − 0.244·67-s + 0.963·69-s − 1.42·71-s + 1.17·73-s − 0.900·79-s + 1/9·81-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1878.08\)
Root analytic conductor: \(43.3368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.28277624294036, −12.54143212095758, −12.24569851217167, −11.74696356550088, −11.32007384147725, −10.76207610522394, −10.41224402183506, −10.00231835848105, −9.364482934865093, −9.187405367667803, −8.239668489650149, −7.888962230068379, −7.438635573602814, −7.159014813687291, −6.383117631843572, −5.797334821070952, −5.496661443296245, −4.993822504231624, −4.559834023635429, −3.883646484076131, −3.238168739128265, −2.722154491064334, −2.020343999610391, −1.618459229924298, −0.5006000358772635, 0, 0.5006000358772635, 1.618459229924298, 2.020343999610391, 2.722154491064334, 3.238168739128265, 3.883646484076131, 4.559834023635429, 4.993822504231624, 5.496661443296245, 5.797334821070952, 6.383117631843572, 7.159014813687291, 7.438635573602814, 7.888962230068379, 8.239668489650149, 9.187405367667803, 9.364482934865093, 10.00231835848105, 10.41224402183506, 10.76207610522394, 11.32007384147725, 11.74696356550088, 12.24569851217167, 12.54143212095758, 13.28277624294036

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