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Properties

Label 2-40e2-16.13-c1-0-6
Degree $2$
Conductor $1600$
Sign $0.871 + 0.491i$
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  + (−2.16 − 2.16i)3-s − 3.30i·7-s + 6.40i·9-s + (−2.01 + 2.01i)11-s + (0.794 + 0.794i)13-s + 4.61·17-s + (3.48 + 3.48i)19-s + (−7.16 + 7.16i)21-s + 7.99i·23-s + (7.38 − 7.38i)27-s + (−1.95 − 1.95i)29-s + 5.12·31-s + 8.72·33-s + (0.448 − 0.448i)37-s − 3.44i·39-s + ⋯
L(s)  = 1  + (−1.25 − 1.25i)3-s − 1.24i·7-s + 2.13i·9-s + (−0.606 + 0.606i)11-s + (0.220 + 0.220i)13-s + 1.11·17-s + (0.800 + 0.800i)19-s + (−1.56 + 1.56i)21-s + 1.66i·23-s + (1.42 − 1.42i)27-s + (−0.362 − 0.362i)29-s + 0.920·31-s + 1.51·33-s + (0.0736 − 0.0736i)37-s − 0.551i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.871 + 0.491i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.871 + 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9611349649\)
\(L(\frac12)\) \(\approx\) \(0.9611349649\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 2.16i)T + 3iT^{2} \)
7 \( 1 + 3.30iT - 7T^{2} \)
11 \( 1 + (2.01 - 2.01i)T - 11iT^{2} \)
13 \( 1 + (-0.794 - 0.794i)T + 13iT^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + (-3.48 - 3.48i)T + 19iT^{2} \)
23 \( 1 - 7.99iT - 23T^{2} \)
29 \( 1 + (1.95 + 1.95i)T + 29iT^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + (-0.448 + 0.448i)T - 37iT^{2} \)
41 \( 1 - 4.02iT - 41T^{2} \)
43 \( 1 + (4.97 - 4.97i)T - 43iT^{2} \)
47 \( 1 - 5.49T + 47T^{2} \)
53 \( 1 + (3.35 - 3.35i)T - 53iT^{2} \)
59 \( 1 + (2.07 - 2.07i)T - 59iT^{2} \)
61 \( 1 + (0.557 + 0.557i)T + 61iT^{2} \)
67 \( 1 + (-0.636 - 0.636i)T + 67iT^{2} \)
71 \( 1 - 6.85iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + (9.48 + 9.48i)T + 83iT^{2} \)
89 \( 1 - 7.62iT - 89T^{2} \)
97 \( 1 - 0.709T + 97T^{2} \)
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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

−9.669496190101951735929346403267, −7.972453141585141676753418233655, −7.60865598261117832698571825430, −7.06698929848291655466921134493, −6.09275565306039232998097977813, −5.45791076710107216694200837449, −4.54149504219965476491663499758, −3.31464377680552234231449705146, −1.69744029528121576328558314942, −0.933627363233207720958346766151, 0.60549631743560578158524955174, 2.67429335834191737091228008334, 3.58200523590811282610677824694, 4.79585617870894625064667372111, 5.35339663490980300026381385471, 5.89469470744560462513445234829, 6.73679563494513912927020949244, 8.092735143885579642383278922255, 8.880853840023461646254132697761, 9.574780573261924569735844040658

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