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Properties

Label 2-40e2-80.27-c1-0-32
Degree $2$
Conductor $1600$
Sign $-0.584 - 0.811i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + (−3 − 3i)7-s + 9-s + (1 − i)11-s − 2i·13-s + (−1 − i)17-s + (3 − 3i)19-s + (6 + 6i)21-s + (−1 + i)23-s + 4·27-s + (−7 − 7i)29-s + 2i·31-s + (−2 + 2i)33-s + 6i·37-s + 4i·39-s + ⋯
L(s)  = 1  − 1.15·3-s + (−1.13 − 1.13i)7-s + 0.333·9-s + (0.301 − 0.301i)11-s − 0.554i·13-s + (−0.242 − 0.242i)17-s + (0.688 − 0.688i)19-s + (1.30 + 1.30i)21-s + (−0.208 + 0.208i)23-s + 0.769·27-s + (−1.29 − 1.29i)29-s + 0.359i·31-s + (−0.348 + 0.348i)33-s + 0.986i·37-s + 0.640i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.584 - 0.811i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 3T^{2} \)
7 \( 1 + (3 + 3i)T + 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (7 + 7i)T + 29iT^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (7 - 7i)T - 47iT^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (3 + 3i)T + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 + 3i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-11 - 11i)T + 97iT^{2} \)
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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

−9.101139535687888154757343634177, −7.896187354212271565040723769402, −7.07867655731667953341167169694, −6.41390294988502804524204609402, −5.74396736749512175495356481205, −4.80692827445366092158236224068, −3.81070009470153480989315449889, −2.90384284904392945037127602770, −0.992545958293620806338961011892, 0, 1.80318515915811479115274602692, 3.07283504649750636870310439265, 4.11119843435503972832553631872, 5.37856479852763345216332433780, 5.75715420349267141210556490114, 6.56719111629516412301280577319, 7.22255523085825466519027665419, 8.532386326098033075536076699371, 9.238942270210354711088330763436

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