Properties

Label 2-3200-8.5-c1-0-5
Degree $2$
Conductor $3200$
Sign $i$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.14i·3-s − 6.89·9-s + 6.61i·11-s − 7.89·17-s − 2.51i·19-s − 12.2i·27-s − 20.7·33-s + 12.7·41-s + 8.48i·43-s − 7·49-s − 24.8i·51-s + 7.89·57-s − 14.1i·59-s + 7.88i·67-s − 13.6·73-s + ⋯
L(s)  = 1  + 1.81i·3-s − 2.29·9-s + 1.99i·11-s − 1.91·17-s − 0.575i·19-s − 2.36i·27-s − 3.62·33-s + 1.99·41-s + 1.29i·43-s − 49-s − 3.48i·51-s + 1.04·57-s − 1.84i·59-s + 0.962i·67-s − 1.60·73-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $i$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5097605613\)
\(L(\frac12)\) \(\approx\) \(0.5097605613\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.14iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 6.61iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.89T + 17T^{2} \)
19 \( 1 + 2.51iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 12.7T + 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 7.88iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.351897823004443386940357212745, −8.845778431296596599617255638951, −7.84830134758384981043218566044, −6.92164348675091749525543909020, −6.12260008916806473155457670815, −4.99921577116159331332355433200, −4.52728873781391495617902336603, −4.13419129508342722091459242759, −2.91413828786457210999861216894, −2.08213856787426906996728661832, 0.16168061828592726358013354107, 1.13753538435333575524224412542, 2.23603266924591935745587948256, 2.96168533417526199989981647943, 4.04611958600484910536531856053, 5.41494446711332131873101037204, 6.11632274523036361135323372487, 6.50856756334546550174555864997, 7.38224003673974474936706578105, 8.042511493470553629528071090085

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