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Properties

Label 2-40e2-5.2-c0-0-2
Degree $2$
Conductor $1600$
Sign $0.850 + 0.525i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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 − i)3-s + (1 + i)7-s − i·9-s + 2·21-s + (−1 + i)23-s − 2i·29-s + (−1 + i)43-s + (−1 − i)47-s + i·49-s + (1 − i)63-s + (−1 − i)67-s + 2i·69-s + 81-s + (−1 + i)83-s + (−2 − 2i)87-s + ⋯
L(s)  = 1  + (1 − i)3-s + (1 + i)7-s − i·9-s + 2·21-s + (−1 + i)23-s − 2i·29-s + (−1 + i)43-s + (−1 − i)47-s + i·49-s + (1 − i)63-s + (−1 − i)67-s + 2i·69-s + 81-s + (−1 + i)83-s + (−2 − 2i)87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.673553564\)
\(L(\frac12)\) \(\approx\) \(1.673553564\)
\(L(1)\) 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 \( 1 + (-1 + i)T - iT^{2} \)
7 \( 1 + (-1 - i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
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

−9.336137767135761875116565430071, −8.424259432951252227075141732133, −8.081474969791895606429605224165, −7.42271215906803915216995867190, −6.35910984698418105891142023115, −5.55617438434589328247878086889, −4.50916375256853110461324208476, −3.27385646867688932235280580080, −2.24492217076563886293446112366, −1.64300569239688915817781434294, 1.59584208834128298357390072734, 2.89458068973806471730627222799, 3.84278008463460074942882744315, 4.48116622592691338547202548855, 5.22812367134743977701325713085, 6.59533025265282169763422979388, 7.49852891830371670949769420491, 8.290612172734234462516375814948, 8.768539091866812237486005072799, 9.702788932087076384253681683392

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