Properties

Label 2-3200-1.1-c1-0-35
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  + 2.90·3-s − 3.52·7-s + 5.42·9-s + 3.80·11-s + 2.62·13-s + 5.80·17-s − 5.05·19-s − 10.2·21-s − 0.474·23-s + 7.05·27-s − 2·29-s + 2.75·31-s + 11.0·33-s + 7.18·37-s + 7.61·39-s + 5.18·41-s − 1.95·43-s + 5.33·47-s + 5.42·49-s + 16.8·51-s + 5.37·53-s − 14.6·57-s + 5.05·59-s − 12.2·61-s − 19.1·63-s − 7.76·67-s − 1.37·69-s + ⋯
L(s)  = 1  + 1.67·3-s − 1.33·7-s + 1.80·9-s + 1.14·11-s + 0.727·13-s + 1.40·17-s − 1.15·19-s − 2.23·21-s − 0.0989·23-s + 1.35·27-s − 0.371·29-s + 0.494·31-s + 1.92·33-s + 1.18·37-s + 1.21·39-s + 0.809·41-s − 0.297·43-s + 0.777·47-s + 0.775·49-s + 2.36·51-s + 0.738·53-s − 1.94·57-s + 0.657·59-s − 1.56·61-s − 2.41·63-s − 0.948·67-s − 0.165·69-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

\(L(1)\) \(\approx\) \(3.458038829\)
\(L(\frac12)\) \(\approx\) \(3.458038829\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.90T + 3T^{2} \)
7 \( 1 + 3.52T + 7T^{2} \)
11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 + 5.05T + 19T^{2} \)
23 \( 1 + 0.474T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 2.75T + 31T^{2} \)
37 \( 1 - 7.18T + 37T^{2} \)
41 \( 1 - 5.18T + 41T^{2} \)
43 \( 1 + 1.95T + 43T^{2} \)
47 \( 1 - 5.33T + 47T^{2} \)
53 \( 1 - 5.37T + 53T^{2} \)
59 \( 1 - 5.05T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 7.76T + 67T^{2} \)
71 \( 1 - 4.85T + 71T^{2} \)
73 \( 1 + 6.66T + 73T^{2} \)
79 \( 1 - 5.24T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−8.893714395259092785365948333817, −7.985965752575430432024303389683, −7.35757565867414757516501283713, −6.43118157873599571573687003388, −5.94878579842365968820477616613, −4.35854511843336284965666949344, −3.70162419160270926244400326940, −3.18477104255624701687650277508, −2.27229583425703584225485956923, −1.10224269720422118375847820065, 1.10224269720422118375847820065, 2.27229583425703584225485956923, 3.18477104255624701687650277508, 3.70162419160270926244400326940, 4.35854511843336284965666949344, 5.94878579842365968820477616613, 6.43118157873599571573687003388, 7.35757565867414757516501283713, 7.985965752575430432024303389683, 8.893714395259092785365948333817

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