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Properties

Label 2-40e2-80.69-c1-0-27
Degree $2$
Conductor $1600$
Sign $0.556 + 0.831i$
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  + (1.09 + 1.09i)3-s + 0.973·7-s − 0.616i·9-s + (−1.40 − 1.40i)11-s + (−4.60 − 4.60i)13-s − 0.490i·17-s + (4.54 − 4.54i)19-s + (1.06 + 1.06i)21-s − 1.94·23-s + (3.94 − 3.94i)27-s + (3.74 − 3.74i)29-s − 4.29·31-s − 3.07i·33-s + (4.55 − 4.55i)37-s − 10.0i·39-s + ⋯
L(s)  = 1  + (0.630 + 0.630i)3-s + 0.368·7-s − 0.205i·9-s + (−0.424 − 0.424i)11-s + (−1.27 − 1.27i)13-s − 0.118i·17-s + (1.04 − 1.04i)19-s + (0.231 + 0.231i)21-s − 0.405·23-s + (0.759 − 0.759i)27-s + (0.695 − 0.695i)29-s − 0.770·31-s − 0.535i·33-s + (0.748 − 0.748i)37-s − 1.60i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.556 + 0.831i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.556 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.777127370\)
\(L(\frac12)\) \(\approx\) \(1.777127370\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 1.09i)T + 3iT^{2} \)
7 \( 1 - 0.973T + 7T^{2} \)
11 \( 1 + (1.40 + 1.40i)T + 11iT^{2} \)
13 \( 1 + (4.60 + 4.60i)T + 13iT^{2} \)
17 \( 1 + 0.490iT - 17T^{2} \)
19 \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \)
23 \( 1 + 1.94T + 23T^{2} \)
29 \( 1 + (-3.74 + 3.74i)T - 29iT^{2} \)
31 \( 1 + 4.29T + 31T^{2} \)
37 \( 1 + (-4.55 + 4.55i)T - 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (1.79 - 1.79i)T - 43iT^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + (5.61 - 5.61i)T - 53iT^{2} \)
59 \( 1 + (-8.44 - 8.44i)T + 59iT^{2} \)
61 \( 1 + (-3.01 + 3.01i)T - 61iT^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 + 0.897iT - 71T^{2} \)
73 \( 1 - 9.71T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + (0.815 + 0.815i)T + 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 - 7.54iT - 97T^{2} \)
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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

−9.434730863040266947073970730791, −8.501314262408677689902135480422, −7.79505123581763293654568480216, −7.10699499772499875669905899447, −5.81828926976821684933389708097, −5.09726310106479433260635482668, −4.22162962379285913674664912531, −3.10287521472293131914901741818, −2.52837570989630046949338472153, −0.62319441979392258487314303178, 1.58911263251858282965084169842, 2.28489893298854757419011071955, 3.39802814568428082395389873451, 4.67135354491055583954086672548, 5.24523417862543320369738618985, 6.59313879912712935595505613632, 7.24148749368924681848408956861, 7.950690575407607606636123212451, 8.494100500168252927320159884250, 9.689150689720454614860532216256

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