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Properties

Label 2-40e2-80.3-c1-0-29
Degree $2$
Conductor $1600$
Sign $-0.809 + 0.586i$
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  + 0.692·3-s + (−0.343 + 0.343i)7-s − 2.52·9-s + (−0.843 − 0.843i)11-s + 3.68i·13-s + (−0.412 + 0.412i)17-s + (−5.37 − 5.37i)19-s + (−0.238 + 0.238i)21-s + (−3.08 − 3.08i)23-s − 3.82·27-s + (4.22 − 4.22i)29-s − 8.75i·31-s + (−0.584 − 0.584i)33-s − 5.41i·37-s + 2.55i·39-s + ⋯
L(s)  = 1  + 0.399·3-s + (−0.129 + 0.129i)7-s − 0.840·9-s + (−0.254 − 0.254i)11-s + 1.02i·13-s + (−0.0999 + 0.0999i)17-s + (−1.23 − 1.23i)19-s + (−0.0519 + 0.0519i)21-s + (−0.643 − 0.643i)23-s − 0.735·27-s + (0.785 − 0.785i)29-s − 1.57i·31-s + (−0.101 − 0.101i)33-s − 0.890i·37-s + 0.408i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.809 + 0.586i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.809 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5050118530\)
\(L(\frac12)\) \(\approx\) \(0.5050118530\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.692T + 3T^{2} \)
7 \( 1 + (0.343 - 0.343i)T - 7iT^{2} \)
11 \( 1 + (0.843 + 0.843i)T + 11iT^{2} \)
13 \( 1 - 3.68iT - 13T^{2} \)
17 \( 1 + (0.412 - 0.412i)T - 17iT^{2} \)
19 \( 1 + (5.37 + 5.37i)T + 19iT^{2} \)
23 \( 1 + (3.08 + 3.08i)T + 23iT^{2} \)
29 \( 1 + (-4.22 + 4.22i)T - 29iT^{2} \)
31 \( 1 + 8.75iT - 31T^{2} \)
37 \( 1 + 5.41iT - 37T^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 - 4.30iT - 43T^{2} \)
47 \( 1 + (4.56 + 4.56i)T + 47iT^{2} \)
53 \( 1 + 6.07T + 53T^{2} \)
59 \( 1 + (7.33 - 7.33i)T - 59iT^{2} \)
61 \( 1 + (4.81 + 4.81i)T + 61iT^{2} \)
67 \( 1 + 14.3iT - 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 + (6.87 - 6.87i)T - 73iT^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 + 1.10T + 89T^{2} \)
97 \( 1 + (7.15 - 7.15i)T - 97iT^{2} \)
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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

−9.077710144219395737625988300157, −8.349724950517127952915760051890, −7.68523545474676077687018598494, −6.44469545407707513094585565093, −6.09640257568633146959130233804, −4.77812657559183366295145651649, −4.05779160106737399145121605037, −2.80004248206774547099056591578, −2.12193917229839139361844151587, −0.16970617694842465671925696869, 1.67144344629563122705737452965, 2.90125086559012091555680160700, 3.59044991600036154154075985722, 4.80076112375197730679605829528, 5.67627912339814145327062337786, 6.45005183826256585183869577633, 7.48057020980718694979191777341, 8.323819333161395019393921066895, 8.652247728882763237748494235293, 9.831212235404897138487210430166

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