Properties

Label 2-3200-1.1-c1-0-3
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  − 1.70·3-s − 2.63·7-s − 0.0783·9-s − 5.41·11-s + 6.34·13-s − 3.41·17-s − 3.26·19-s + 4.49·21-s − 1.36·23-s + 5.26·27-s − 2·29-s − 4.68·31-s + 9.26·33-s − 5.75·37-s − 10.8·39-s − 7.75·41-s + 4.44·43-s − 4.78·47-s − 0.0783·49-s + 5.84·51-s + 1.65·53-s + 5.57·57-s + 3.26·59-s + 2.49·61-s + 0.206·63-s + 7.86·67-s + 2.34·69-s + ⋯
L(s)  = 1  − 0.986·3-s − 0.994·7-s − 0.0261·9-s − 1.63·11-s + 1.75·13-s − 0.829·17-s − 0.748·19-s + 0.981·21-s − 0.285·23-s + 1.01·27-s − 0.371·29-s − 0.840·31-s + 1.61·33-s − 0.946·37-s − 1.73·39-s − 1.21·41-s + 0.678·43-s − 0.698·47-s − 0.0111·49-s + 0.818·51-s + 0.227·53-s + 0.738·57-s + 0.424·59-s + 0.319·61-s + 0.0259·63-s + 0.960·67-s + 0.281·69-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

\(L(1)\) \(\approx\) \(0.4957691566\)
\(L(\frac12)\) \(\approx\) \(0.4957691566\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.70T + 3T^{2} \)
7 \( 1 + 2.63T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 6.34T + 13T^{2} \)
17 \( 1 + 3.41T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 + 1.36T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 + 7.75T + 41T^{2} \)
43 \( 1 - 4.44T + 43T^{2} \)
47 \( 1 + 4.78T + 47T^{2} \)
53 \( 1 - 1.65T + 53T^{2} \)
59 \( 1 - 3.26T + 59T^{2} \)
61 \( 1 - 2.49T + 61T^{2} \)
67 \( 1 - 7.86T + 67T^{2} \)
71 \( 1 + 6.15T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 8.52T + 89T^{2} \)
97 \( 1 - 4.58T + 97T^{2} \)
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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

−8.575095958067233187215847303373, −8.053756359180296662291231431697, −6.82075854402774242069948601881, −6.41657002393830931539206937117, −5.63590454449603061374276801229, −5.10226577404750916164069958989, −3.93492390988198857089776231771, −3.14354575400268931518202721820, −2.02729712150983965517664289534, −0.42084165455400497960245270812, 0.42084165455400497960245270812, 2.02729712150983965517664289534, 3.14354575400268931518202721820, 3.93492390988198857089776231771, 5.10226577404750916164069958989, 5.63590454449603061374276801229, 6.41657002393830931539206937117, 6.82075854402774242069948601881, 8.053756359180296662291231431697, 8.575095958067233187215847303373

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