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Properties

Label 2-40e2-40.27-c1-0-32
Degree $2$
Conductor $1600$
Sign $-0.287 + 0.957i$
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  + (2.18 − 2.18i)3-s + (1.79 − 1.79i)7-s − 6.58i·9-s + 0.913·11-s + (1.73 + 1.73i)13-s + (−3 − 3i)17-s − 3.46i·19-s − 7.84i·21-s + (3.79 + 3.79i)23-s + (−7.84 − 7.84i)27-s − 5.29·29-s + 7.58i·31-s + (1.99 − 1.99i)33-s + (5.19 − 5.19i)37-s + 7.58·39-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)3-s + (0.677 − 0.677i)7-s − 2.19i·9-s + 0.275·11-s + (0.480 + 0.480i)13-s + (−0.727 − 0.727i)17-s − 0.794i·19-s − 1.71i·21-s + (0.790 + 0.790i)23-s + (−1.50 − 1.50i)27-s − 0.982·29-s + 1.36i·31-s + (0.348 − 0.348i)33-s + (0.854 − 0.854i)37-s + 1.21·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.868164384\)
\(L(\frac12)\) \(\approx\) \(2.868164384\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 + 2.18i)T - 3iT^{2} \)
7 \( 1 + (-1.79 + 1.79i)T - 7iT^{2} \)
11 \( 1 - 0.913T + 11T^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-3.79 - 3.79i)T + 23iT^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 - 7.58iT - 31T^{2} \)
37 \( 1 + (-5.19 + 5.19i)T - 37iT^{2} \)
41 \( 1 - 1.58T + 41T^{2} \)
43 \( 1 + (-0.361 + 0.361i)T - 43iT^{2} \)
47 \( 1 + (3.79 - 3.79i)T - 47iT^{2} \)
53 \( 1 + (2.64 + 2.64i)T + 53iT^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 - 6.20iT - 61T^{2} \)
67 \( 1 + (0.361 + 0.361i)T + 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (-8.58 + 8.58i)T - 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (-8.58 - 8.58i)T + 97iT^{2} \)
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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

−9.047887467527364354717517537394, −8.350397116458204963935080707390, −7.42267162929744798968191134301, −7.12969951251301641950799995991, −6.28258255032521217927827806918, −4.91246552585863861457353927675, −3.87624389375403587567356827819, −2.93479617233850474577153401865, −1.89463147961070180141042112240, −1.01126398101893894452848473263, 1.85189446431274455332675519526, 2.76925212149475013949940739682, 3.76615661926936189899057247371, 4.43147360311996142734389183435, 5.33319211759719402759723535021, 6.27407219759564172198290413269, 7.69740383401877930991964260047, 8.300993060578695678868335741768, 8.819668717890421024221313111529, 9.509635117011834504437927374160

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