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Properties

Label 2-40e2-40.13-c0-0-0
Degree $2$
Conductor $1600$
Sign $-0.991 + 0.130i$
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  + (−1.22 + 1.22i)3-s − 1.99i·9-s + i·11-s + (−1.22 + 1.22i)17-s − 19-s + (1.22 + 1.22i)27-s + (−1.22 − 1.22i)33-s − 41-s + i·49-s − 2.99i·51-s + (1.22 − 1.22i)57-s − 2·59-s + (−1.22 − 1.22i)67-s + (−1.22 − 1.22i)73-s − 0.999·81-s + ⋯
L(s)  = 1  + (−1.22 + 1.22i)3-s − 1.99i·9-s + i·11-s + (−1.22 + 1.22i)17-s − 19-s + (1.22 + 1.22i)27-s + (−1.22 − 1.22i)33-s − 41-s + i·49-s − 2.99i·51-s + (1.22 − 1.22i)57-s − 2·59-s + (−1.22 − 1.22i)67-s + (−1.22 − 1.22i)73-s − 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.991 + 0.130i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ -0.991 + 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3526370335\)
\(L(\frac12)\) \(\approx\) \(0.3526370335\)
\(L(1)\) 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 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
89 \( 1 - iT - T^{2} \)
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

−10.30386635226163498227395604186, −9.332497173896699128256980730011, −8.733792681194620432605975176290, −7.53303545394306369699808766892, −6.37475380718178437349379253851, −6.08920768080841856083918899542, −4.73436185053315994444534527764, −4.54605955581184420751376786667, −3.52831827802221885680854809841, −1.91812195719721418456131785844, 0.31114918539938107097133791348, 1.71581791989755751048244052630, 2.84813675121973321726684477036, 4.38999806617386669448990219108, 5.28394355768444367825233753659, 6.07622301929319422954706774096, 6.71156282822601169548260586302, 7.32662185321709633231454750380, 8.311208686279505299341235181324, 8.993541772099982841955600152602

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