Properties

Label 2-3200-1.1-c1-0-9
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  − 2.41·3-s − 0.828·7-s + 2.82·9-s + 5.24·11-s − 5.65·13-s + 0.171·17-s + 1.58·19-s + 1.99·21-s + 4.82·23-s + 0.414·27-s − 8·29-s − 0.828·31-s − 12.6·33-s − 7.65·37-s + 13.6·39-s + 10.6·41-s − 10·43-s + 9.65·47-s − 6.31·49-s − 0.414·51-s + 7.65·53-s − 3.82·57-s − 3.65·59-s − 6·61-s − 2.34·63-s + 2.75·67-s − 11.6·69-s + ⋯
L(s)  = 1  − 1.39·3-s − 0.313·7-s + 0.942·9-s + 1.58·11-s − 1.56·13-s + 0.0416·17-s + 0.363·19-s + 0.436·21-s + 1.00·23-s + 0.0797·27-s − 1.48·29-s − 0.148·31-s − 2.20·33-s − 1.25·37-s + 2.18·39-s + 1.66·41-s − 1.52·43-s + 1.40·47-s − 0.901·49-s − 0.0580·51-s + 1.05·53-s − 0.507·57-s − 0.476·59-s − 0.768·61-s − 0.295·63-s + 0.336·67-s − 1.40·69-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

\(L(1)\) \(\approx\) \(0.8719056306\)
\(L(\frac12)\) \(\approx\) \(0.8719056306\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 3T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 - 5.24T + 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 0.171T + 17T^{2} \)
19 \( 1 - 1.58T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + 0.828T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 - 9.65T + 47T^{2} \)
53 \( 1 - 7.65T + 53T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 2.75T + 67T^{2} \)
71 \( 1 + 9.65T + 71T^{2} \)
73 \( 1 + 5.82T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + 1.24T + 83T^{2} \)
89 \( 1 - 6.17T + 89T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−9.004226813107531565084218041729, −7.56887103064768595205763100330, −7.03676014516620699102890183706, −6.41030030655787428061247477010, −5.60792138824045654886823473972, −4.97880710637258612649134856336, −4.17038576256479047738542312157, −3.15978815668361773663467847378, −1.80009446496141328112468694990, −0.60600542720599434226467620275, 0.60600542720599434226467620275, 1.80009446496141328112468694990, 3.15978815668361773663467847378, 4.17038576256479047738542312157, 4.97880710637258612649134856336, 5.60792138824045654886823473972, 6.41030030655787428061247477010, 7.03676014516620699102890183706, 7.56887103064768595205763100330, 9.004226813107531565084218041729

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