Properties

Label 2-40e2-1.1-c3-0-107
Degree $2$
Conductor $1600$
Sign $-1$
Analytic cond. $94.4030$
Root an. cond. $9.71612$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.47·3-s + 31.3·7-s − 6.99·9-s + 8.94·11-s − 62·13-s + 46·17-s − 107.·19-s + 140·21-s − 192.·23-s − 152.·27-s + 90·29-s − 152.·31-s + 40.0·33-s − 214·37-s − 277.·39-s − 10·41-s − 67.0·43-s − 398.·47-s + 637.·49-s + 205.·51-s − 678·53-s − 480.·57-s + 411.·59-s − 250·61-s − 219.·63-s + 49.1·67-s − 860·69-s + ⋯
L(s)  = 1  + 0.860·3-s + 1.69·7-s − 0.259·9-s + 0.245·11-s − 1.32·13-s + 0.656·17-s − 1.29·19-s + 1.45·21-s − 1.74·23-s − 1.08·27-s + 0.576·29-s − 0.880·31-s + 0.211·33-s − 0.950·37-s − 1.13·39-s − 0.0380·41-s − 0.237·43-s − 1.23·47-s + 1.85·49-s + 0.564·51-s − 1.75·53-s − 1.11·57-s + 0.907·59-s − 0.524·61-s − 0.438·63-s + 0.0897·67-s − 1.50·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(94.4030\)
Root analytic conductor: \(9.71612\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good3 \( 1 - 4.47T + 27T^{2} \)
7 \( 1 - 31.3T + 343T^{2} \)
11 \( 1 - 8.94T + 1.33e3T^{2} \)
13 \( 1 + 62T + 2.19e3T^{2} \)
17 \( 1 - 46T + 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 + 192.T + 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 + 152.T + 2.97e4T^{2} \)
37 \( 1 + 214T + 5.06e4T^{2} \)
41 \( 1 + 10T + 6.89e4T^{2} \)
43 \( 1 + 67.0T + 7.95e4T^{2} \)
47 \( 1 + 398.T + 1.03e5T^{2} \)
53 \( 1 + 678T + 1.48e5T^{2} \)
59 \( 1 - 411.T + 2.05e5T^{2} \)
61 \( 1 + 250T + 2.26e5T^{2} \)
67 \( 1 - 49.1T + 3.00e5T^{2} \)
71 \( 1 + 366.T + 3.57e5T^{2} \)
73 \( 1 + 522T + 3.89e5T^{2} \)
79 \( 1 - 876.T + 4.93e5T^{2} \)
83 \( 1 - 380.T + 5.71e5T^{2} \)
89 \( 1 - 970T + 7.04e5T^{2} \)
97 \( 1 - 934T + 9.12e5T^{2} \)
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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

−8.422777772779364027269422811470, −8.029291160512341703273170685915, −7.38544009129816468016909387691, −6.18660708472256330599149771780, −5.16460525335862914295096002847, −4.47982670025021152201129192781, −3.47495529244211035028379437294, −2.23769358432621297540510257057, −1.74296813240895893521529564963, 0, 1.74296813240895893521529564963, 2.23769358432621297540510257057, 3.47495529244211035028379437294, 4.47982670025021152201129192781, 5.16460525335862914295096002847, 6.18660708472256330599149771780, 7.38544009129816468016909387691, 8.029291160512341703273170685915, 8.422777772779364027269422811470

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