Properties

Label 2-40e2-20.3-c1-0-31
Degree $2$
Conductor $1600$
Sign $-0.850 + 0.525i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)3-s + (−2 − 2i)7-s − 5i·9-s + (−1 − i)13-s + (5 − 5i)17-s − 4·19-s − 8·21-s + (−2 + 2i)23-s + (−4 − 4i)27-s − 4i·29-s + 4i·31-s + (1 − i)37-s − 4·39-s + (−6 + 6i)43-s + (2 + 2i)47-s + ⋯
L(s)  = 1  + (1.15 − 1.15i)3-s + (−0.755 − 0.755i)7-s − 1.66i·9-s + (−0.277 − 0.277i)13-s + (1.21 − 1.21i)17-s − 0.917·19-s − 1.74·21-s + (−0.417 + 0.417i)23-s + (−0.769 − 0.769i)27-s − 0.742i·29-s + 0.718i·31-s + (0.164 − 0.164i)37-s − 0.640·39-s + (−0.914 + 0.914i)43-s + (0.291 + 0.291i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934604528\)
\(L(\frac12)\) \(\approx\) \(1.934604528\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-2 + 2i)T - 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (6 - 6i)T - 43iT^{2} \)
47 \( 1 + (-2 - 2i)T + 47iT^{2} \)
53 \( 1 + (7 + 7i)T + 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + (10 + 10i)T + 67iT^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + (2 - 2i)T - 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−9.056634582381015403254671607378, −7.977162426659782429432604116678, −7.66347603819499225281650247792, −6.82835382930067116706614921599, −6.18644732074603755814908934176, −4.86483868430625068173221651437, −3.55413398289087669965633427385, −2.99841847909133225820363591973, −1.88358072592583766918140647479, −0.61623069282738966788727087392, 2.01273877255625461573063174726, 2.97129237650366007108472251230, 3.71583613784677420748047545493, 4.50453359516864159349933091629, 5.59966231196990228066846491567, 6.39502234215916413787621248104, 7.62156143975848594067059768836, 8.477534479365718684976487729233, 8.890910436929955606763598500602, 9.760988051984675591341229533649

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