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Properties

Label 2-40e2-80.3-c1-0-19
Degree $2$
Conductor $1600$
Sign $0.659 - 0.751i$
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  + 3.25·3-s + (−2.54 + 2.54i)7-s + 7.61·9-s + (0.462 + 0.462i)11-s + 1.33i·13-s + (−2.37 + 2.37i)17-s + (2.69 + 2.69i)19-s + (−8.27 + 8.27i)21-s + (2.10 + 2.10i)23-s + 15.0·27-s + (1.97 − 1.97i)29-s + 7.03i·31-s + (1.50 + 1.50i)33-s − 7.81i·37-s + 4.34i·39-s + ⋯
L(s)  = 1  + 1.88·3-s + (−0.960 + 0.960i)7-s + 2.53·9-s + (0.139 + 0.139i)11-s + 0.370i·13-s + (−0.575 + 0.575i)17-s + (0.618 + 0.618i)19-s + (−1.80 + 1.80i)21-s + (0.438 + 0.438i)23-s + 2.89·27-s + (0.367 − 0.367i)29-s + 1.26i·31-s + (0.262 + 0.262i)33-s − 1.28i·37-s + 0.696i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.095315013\)
\(L(\frac12)\) \(\approx\) \(3.095315013\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.25T + 3T^{2} \)
7 \( 1 + (2.54 - 2.54i)T - 7iT^{2} \)
11 \( 1 + (-0.462 - 0.462i)T + 11iT^{2} \)
13 \( 1 - 1.33iT - 13T^{2} \)
17 \( 1 + (2.37 - 2.37i)T - 17iT^{2} \)
19 \( 1 + (-2.69 - 2.69i)T + 19iT^{2} \)
23 \( 1 + (-2.10 - 2.10i)T + 23iT^{2} \)
29 \( 1 + (-1.97 + 1.97i)T - 29iT^{2} \)
31 \( 1 - 7.03iT - 31T^{2} \)
37 \( 1 + 7.81iT - 37T^{2} \)
41 \( 1 + 2.17iT - 41T^{2} \)
43 \( 1 + 3.10iT - 43T^{2} \)
47 \( 1 + (0.0727 + 0.0727i)T + 47iT^{2} \)
53 \( 1 + 0.719T + 53T^{2} \)
59 \( 1 + (-8.67 + 8.67i)T - 59iT^{2} \)
61 \( 1 + (7.10 + 7.10i)T + 61iT^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-0.905 + 0.905i)T - 73iT^{2} \)
79 \( 1 + 3.90T + 79T^{2} \)
83 \( 1 + 6.02T + 83T^{2} \)
89 \( 1 + 7.46T + 89T^{2} \)
97 \( 1 + (3.74 - 3.74i)T - 97iT^{2} \)
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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

−9.412709469417052243779239502615, −8.749621904599762981801171728564, −8.195732608404519156151480245293, −7.20115998134154839313817807116, −6.55888319869924930379184605889, −5.38466058989421285600464549061, −4.09594725730860287408397141372, −3.39098314572866647020694657233, −2.58488343808646049887508877061, −1.70446031287955533651120932502, 1.00508376787031812542414575073, 2.54003883982012965750975971112, 3.14885981699264826276198256477, 3.96785253824889521634329456287, 4.80837162547630514519961614600, 6.44664082219226391753207759193, 7.10271955122697743572912178043, 7.74030222334838137924983837791, 8.571265259874831822616409315659, 9.299380168150577522591353370896

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