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Properties

Label 2-40e2-80.67-c1-0-6
Degree $2$
Conductor $1600$
Sign $-0.507 - 0.861i$
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  + 2.02i·3-s + (3.26 + 3.26i)7-s − 1.11·9-s + (−3.55 − 3.55i)11-s + 1.41·13-s + (1.73 + 1.73i)17-s + (4.19 + 4.19i)19-s + (−6.63 + 6.63i)21-s + (−0.177 + 0.177i)23-s + 3.82i·27-s + (1.63 − 1.63i)29-s + 8.15i·31-s + (7.20 − 7.20i)33-s − 1.34·37-s + 2.87i·39-s + ⋯
L(s)  = 1  + 1.17i·3-s + (1.23 + 1.23i)7-s − 0.371·9-s + (−1.07 − 1.07i)11-s + 0.393·13-s + (0.420 + 0.420i)17-s + (0.962 + 0.962i)19-s + (−1.44 + 1.44i)21-s + (−0.0370 + 0.0370i)23-s + 0.735i·27-s + (0.304 − 0.304i)29-s + 1.46i·31-s + (1.25 − 1.25i)33-s − 0.220·37-s + 0.460i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.897956744\)
\(L(\frac12)\) \(\approx\) \(1.897956744\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.02iT - 3T^{2} \)
7 \( 1 + (-3.26 - 3.26i)T + 7iT^{2} \)
11 \( 1 + (3.55 + 3.55i)T + 11iT^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
17 \( 1 + (-1.73 - 1.73i)T + 17iT^{2} \)
19 \( 1 + (-4.19 - 4.19i)T + 19iT^{2} \)
23 \( 1 + (0.177 - 0.177i)T - 23iT^{2} \)
29 \( 1 + (-1.63 + 1.63i)T - 29iT^{2} \)
31 \( 1 - 8.15iT - 31T^{2} \)
37 \( 1 + 1.34T + 37T^{2} \)
41 \( 1 - 1.16iT - 41T^{2} \)
43 \( 1 + 1.04T + 43T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 + 8.19iT - 53T^{2} \)
59 \( 1 + (-3.96 + 3.96i)T - 59iT^{2} \)
61 \( 1 + (7.22 + 7.22i)T + 61iT^{2} \)
67 \( 1 + 9.19T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-5.86 - 5.86i)T + 73iT^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-5.71 - 5.71i)T + 97iT^{2} \)
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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

−9.744545563376116994796986388686, −8.707419938817143512890022528338, −8.390606837983175789965200125336, −7.56731395977978202850992621169, −6.05207611594834496272419619366, −5.33825750331574603777635533309, −4.95591603842182386301485487738, −3.71406610159910288779513027408, −2.91438355046621955348343228650, −1.55744679585442852754218803022, 0.793626452737198888656375514496, 1.71043868578775598911797824479, 2.77649311590143874334573985990, 4.29284731312041076823780370390, 4.89806944052344321106323666612, 5.96537699180113441147387212648, 7.26294240577428497278592626662, 7.39243077729508168852300105871, 7.908926417507303079541169955920, 9.016572617543874618808650477395

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