Properties

Degree $2$
Conductor $78400$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·11-s + 6·13-s + 2·17-s − 6·29-s + 8·31-s − 10·37-s − 2·41-s − 4·43-s − 8·47-s − 2·53-s + 8·59-s − 14·61-s + 12·67-s + 16·71-s + 2·73-s + 8·79-s + 9·81-s + 8·83-s − 10·89-s + 2·97-s − 12·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 9-s + 1.20·11-s + 1.66·13-s + 0.485·17-s − 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 1.16·47-s − 0.274·53-s + 1.04·59-s − 1.79·61-s + 1.46·67-s + 1.89·71-s + 0.234·73-s + 0.900·79-s + 81-s + 0.878·83-s − 1.05·89-s + 0.203·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{78400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 78400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.15261317013702, −13.72921282200575, −13.53959594953131, −12.73700796456256, −12.14136868268929, −11.80031341234956, −11.23298647485335, −10.93891726319854, −10.34181183907830, −9.534514963543027, −9.320333360962897, −8.540886424971603, −8.328678357866788, −7.843625621777835, −6.798596570254665, −6.591447794800471, −6.070406833105631, −5.424421711913087, −4.992896332794901, −4.052443116353610, −3.565057754514165, −3.273179766048373, −2.331265831553527, −1.533842316951171, −1.035004774044961, 0, 1.035004774044961, 1.533842316951171, 2.331265831553527, 3.273179766048373, 3.565057754514165, 4.052443116353610, 4.992896332794901, 5.424421711913087, 6.070406833105631, 6.591447794800471, 6.798596570254665, 7.843625621777835, 8.328678357866788, 8.540886424971603, 9.320333360962897, 9.534514963543027, 10.34181183907830, 10.93891726319854, 11.23298647485335, 11.80031341234956, 12.14136868268929, 12.73700796456256, 13.53959594953131, 13.72921282200575, 14.15261317013702

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