Properties

Label 2-3200-1.1-c1-0-30
Degree $2$
Conductor $3200$
Sign $1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.414·3-s + 4.82·7-s − 2.82·9-s − 3.24·11-s + 5.65·13-s + 5.82·17-s + 4.41·19-s + 1.99·21-s − 0.828·23-s − 2.41·27-s − 8·29-s + 4.82·31-s − 1.34·33-s + 3.65·37-s + 2.34·39-s − 0.656·41-s − 10·43-s − 1.65·47-s + 16.3·49-s + 2.41·51-s − 3.65·53-s + 1.82·57-s + 7.65·59-s − 6·61-s − 13.6·63-s + 11.2·67-s − 0.343·69-s + ⋯
L(s)  = 1  + 0.239·3-s + 1.82·7-s − 0.942·9-s − 0.977·11-s + 1.56·13-s + 1.41·17-s + 1.01·19-s + 0.436·21-s − 0.172·23-s − 0.464·27-s − 1.48·29-s + 0.867·31-s − 0.233·33-s + 0.601·37-s + 0.375·39-s − 0.102·41-s − 1.52·43-s − 0.241·47-s + 2.33·49-s + 0.338·51-s − 0.502·53-s + 0.242·57-s + 0.996·59-s − 0.768·61-s − 1.72·63-s + 1.37·67-s − 0.0413·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.574962004\)
\(L(\frac12)\) \(\approx\) \(2.574962004\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 0.414T + 3T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 5.82T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 - 4.82T + 31T^{2} \)
37 \( 1 - 3.65T + 37T^{2} \)
41 \( 1 + 0.656T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + 3.65T + 53T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 1.65T + 71T^{2} \)
73 \( 1 + 0.171T + 73T^{2} \)
79 \( 1 - 7.17T + 79T^{2} \)
83 \( 1 - 7.24T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 5.31T + 97T^{2} \)
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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

−8.387793127411064329101240827808, −7.999074365969824494290242841104, −7.59489059071052323706659983290, −6.24994609108507013909849993427, −5.35221018591912918383327415071, −5.15976134122521946423128357563, −3.85383871969369583013552300199, −3.10073598257651528713765136961, −1.96881282638433772598652182865, −1.02720602017096475216196914316, 1.02720602017096475216196914316, 1.96881282638433772598652182865, 3.10073598257651528713765136961, 3.85383871969369583013552300199, 5.15976134122521946423128357563, 5.35221018591912918383327415071, 6.24994609108507013909849993427, 7.59489059071052323706659983290, 7.999074365969824494290242841104, 8.387793127411064329101240827808

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