Properties

Label 2-3856-1.1-c1-0-80
Degree $2$
Conductor $3856$
Sign $1$
Analytic cond. $30.7903$
Root an. cond. $5.54890$
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  + 2.45·3-s + 2.74·5-s + 0.283·7-s + 3.00·9-s + 4.12·11-s + 0.0271·13-s + 6.71·15-s − 1.28·17-s + 5.72·19-s + 0.695·21-s + 5.97·23-s + 2.51·25-s + 0.0132·27-s − 2.55·29-s + 2.02·31-s + 10.1·33-s + 0.777·35-s + 2.42·37-s + 0.0666·39-s − 11.0·41-s − 10.4·43-s + 8.24·45-s − 4.54·47-s − 6.91·49-s − 3.15·51-s − 9.30·53-s + 11.3·55-s + ⋯
L(s)  = 1  + 1.41·3-s + 1.22·5-s + 0.107·7-s + 1.00·9-s + 1.24·11-s + 0.00754·13-s + 1.73·15-s − 0.312·17-s + 1.31·19-s + 0.151·21-s + 1.24·23-s + 0.503·25-s + 0.00254·27-s − 0.474·29-s + 0.364·31-s + 1.76·33-s + 0.131·35-s + 0.399·37-s + 0.0106·39-s − 1.72·41-s − 1.60·43-s + 1.22·45-s − 0.662·47-s − 0.988·49-s − 0.441·51-s − 1.27·53-s + 1.52·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3856\)    =    \(2^{4} \cdot 241\)
Sign: $1$
Analytic conductor: \(30.7903\)
Root analytic conductor: \(5.54890\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3856,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.573377107\)
\(L(\frac12)\) \(\approx\) \(4.573377107\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
241 \( 1 + T \)
good3 \( 1 - 2.45T + 3T^{2} \)
5 \( 1 - 2.74T + 5T^{2} \)
7 \( 1 - 0.283T + 7T^{2} \)
11 \( 1 - 4.12T + 11T^{2} \)
13 \( 1 - 0.0271T + 13T^{2} \)
17 \( 1 + 1.28T + 17T^{2} \)
19 \( 1 - 5.72T + 19T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 + 2.55T + 29T^{2} \)
31 \( 1 - 2.02T + 31T^{2} \)
37 \( 1 - 2.42T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 + 4.54T + 47T^{2} \)
53 \( 1 + 9.30T + 53T^{2} \)
59 \( 1 - 9.94T + 59T^{2} \)
61 \( 1 - 8.17T + 61T^{2} \)
67 \( 1 + 4.40T + 67T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 6.69T + 79T^{2} \)
83 \( 1 - 4.32T + 83T^{2} \)
89 \( 1 - 0.746T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.704002151935151580773315836280, −7.893113182094629940308779370867, −6.98700130116792239690493227485, −6.45910956743172460121184600758, −5.45476753566491114120678265571, −4.68599029193326419617218622776, −3.51404785888566711030965745770, −3.04423338321198385771390413309, −1.94900289552784900309013278949, −1.36141145079031845849197045658, 1.36141145079031845849197045658, 1.94900289552784900309013278949, 3.04423338321198385771390413309, 3.51404785888566711030965745770, 4.68599029193326419617218622776, 5.45476753566491114120678265571, 6.45910956743172460121184600758, 6.98700130116792239690493227485, 7.893113182094629940308779370867, 8.704002151935151580773315836280

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