Properties

Label 2-100800-1.1-c1-0-291
Degree $2$
Conductor $100800$
Sign $-1$
Analytic cond. $804.892$
Root an. cond. $28.3706$
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  − 7-s + 2·13-s − 2·19-s + 6·29-s + 8·31-s − 4·37-s − 6·41-s + 2·43-s − 6·47-s + 49-s − 6·53-s + 12·59-s − 8·61-s + 2·67-s − 6·71-s − 2·73-s − 16·79-s − 6·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.554·13-s − 0.458·19-s + 1.11·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s − 0.824·53-s + 1.56·59-s − 1.02·61-s + 0.244·67-s − 0.712·71-s − 0.234·73-s − 1.80·79-s − 0.635·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100800\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(804.892\)
Root analytic conductor: \(28.3706\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.00725579998869, −13.51140623494615, −12.95876184545964, −12.66411703613028, −11.87603735376512, −11.72121685942583, −11.04336384291921, −10.46788099879211, −10.05123650091518, −9.714818520700761, −8.872230556591427, −8.532009991768569, −8.179772166475960, −7.396983070128892, −6.910585028489346, −6.309775989675187, −6.058914696149551, −5.273492371317213, −4.654316572929602, −4.259376137497891, −3.418050196978155, −3.058804669380094, −2.333044022710785, −1.580846174834889, −0.8856363250138436, 0, 0.8856363250138436, 1.580846174834889, 2.333044022710785, 3.058804669380094, 3.418050196978155, 4.259376137497891, 4.654316572929602, 5.273492371317213, 6.058914696149551, 6.309775989675187, 6.910585028489346, 7.396983070128892, 8.179772166475960, 8.532009991768569, 8.872230556591427, 9.714818520700761, 10.05123650091518, 10.46788099879211, 11.04336384291921, 11.72121685942583, 11.87603735376512, 12.66411703613028, 12.95876184545964, 13.51140623494615, 14.00725579998869

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