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Properties

Label 2-3200-40.13-c0-0-9
Degree $2$
Conductor $3200$
Sign $-0.229 + 0.973i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 1.41i)3-s − 3.00i·9-s + (−2.82 − 2.82i)27-s + 2·41-s + (−1.41 + 1.41i)43-s + i·49-s + (−1.41 − 1.41i)67-s − 5.00·81-s + (1.41 − 1.41i)83-s + 2i·89-s + (1.41 + 1.41i)107-s + ⋯
L(s)  = 1  + (1.41 − 1.41i)3-s − 3.00i·9-s + (−2.82 − 2.82i)27-s + 2·41-s + (−1.41 + 1.41i)43-s + i·49-s + (−1.41 − 1.41i)67-s − 5.00·81-s + (1.41 − 1.41i)83-s + 2i·89-s + (1.41 + 1.41i)107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.990403125\)
\(L(\frac12)\) \(\approx\) \(1.990403125\)
\(L(1)\) 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 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.505607149693686386467624107209, −7.78787987516365176361616460642, −7.42568930993858421485916974400, −6.47157746428268670605342186128, −6.02478721038534357347585070934, −4.60768696808172635774657344279, −3.57806610965051525441759092108, −2.86704505575921986834724822173, −2.03682074274106870312811269599, −1.07049165066181939966621197221, 1.92402761349693846825292498693, 2.78110614280661601971547883332, 3.57647308752761538821328187365, 4.25296311854019595907715799349, 4.99574512928044212912366287594, 5.79280762056486374395754416582, 7.11535824710884262847275179906, 7.78523385196533417337525623002, 8.584333175894431631770063746548, 8.955192298152842810323709701742

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