Properties

Label 2-3201-1.1-c1-0-14
Degree $2$
Conductor $3201$
Sign $1$
Analytic cond. $25.5601$
Root an. cond. $5.05570$
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  − 0.860·2-s − 3-s − 1.25·4-s − 1.02·5-s + 0.860·6-s − 2.10·7-s + 2.80·8-s + 9-s + 0.882·10-s − 11-s + 1.25·12-s + 0.448·13-s + 1.81·14-s + 1.02·15-s + 0.105·16-s + 6.98·17-s − 0.860·18-s + 2.35·19-s + 1.29·20-s + 2.10·21-s + 0.860·22-s − 9.42·23-s − 2.80·24-s − 3.94·25-s − 0.385·26-s − 27-s + 2.64·28-s + ⋯
L(s)  = 1  − 0.608·2-s − 0.577·3-s − 0.629·4-s − 0.458·5-s + 0.351·6-s − 0.795·7-s + 0.991·8-s + 0.333·9-s + 0.279·10-s − 0.301·11-s + 0.363·12-s + 0.124·13-s + 0.483·14-s + 0.264·15-s + 0.0263·16-s + 1.69·17-s − 0.202·18-s + 0.540·19-s + 0.288·20-s + 0.459·21-s + 0.183·22-s − 1.96·23-s − 0.572·24-s − 0.789·25-s − 0.0756·26-s − 0.192·27-s + 0.500·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3201\)    =    \(3 \cdot 11 \cdot 97\)
Sign: $1$
Analytic conductor: \(25.5601\)
Root analytic conductor: \(5.05570\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3201,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4523246635\)
\(L(\frac12)\) \(\approx\) \(0.4523246635\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + T \)
97 \( 1 - T \)
good2 \( 1 + 0.860T + 2T^{2} \)
5 \( 1 + 1.02T + 5T^{2} \)
7 \( 1 + 2.10T + 7T^{2} \)
13 \( 1 - 0.448T + 13T^{2} \)
17 \( 1 - 6.98T + 17T^{2} \)
19 \( 1 - 2.35T + 19T^{2} \)
23 \( 1 + 9.42T + 23T^{2} \)
29 \( 1 - 1.98T + 29T^{2} \)
31 \( 1 - 5.95T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 + 8.78T + 41T^{2} \)
43 \( 1 + 9.11T + 43T^{2} \)
47 \( 1 + 3.78T + 47T^{2} \)
53 \( 1 - 9.78T + 53T^{2} \)
59 \( 1 - 6.14T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 1.63T + 83T^{2} \)
89 \( 1 + 2.24T + 89T^{2} \)
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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

−8.492369093712464063808885335441, −8.042474380417479516588219196516, −7.36228743183366837178386650519, −6.42520603200996393101316043237, −5.61013781708330819507319893433, −4.91607970476351571069566157822, −3.88008920186257163335068445057, −3.29873487247235224668779989049, −1.67501091904341700344880173333, −0.46756935762397751828706837141, 0.46756935762397751828706837141, 1.67501091904341700344880173333, 3.29873487247235224668779989049, 3.88008920186257163335068445057, 4.91607970476351571069566157822, 5.61013781708330819507319893433, 6.42520603200996393101316043237, 7.36228743183366837178386650519, 8.042474380417479516588219196516, 8.492369093712464063808885335441

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