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Properties

Label 2-40e2-80.27-c1-0-29
Degree $2$
Conductor $1600$
Sign $-0.812 - 0.583i$
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  − 2.35·3-s + (−2.66 − 2.66i)7-s + 2.56·9-s + (2.20 − 2.20i)11-s − 4.16i·13-s + (−1.69 − 1.69i)17-s + (−4.74 + 4.74i)19-s + (6.28 + 6.28i)21-s + (3.70 − 3.70i)23-s + 1.03·27-s + (3.65 + 3.65i)29-s − 6.90i·31-s + (−5.20 + 5.20i)33-s − 1.10i·37-s + 9.81i·39-s + ⋯
L(s)  = 1  − 1.36·3-s + (−1.00 − 1.00i)7-s + 0.853·9-s + (0.665 − 0.665i)11-s − 1.15i·13-s + (−0.410 − 0.410i)17-s + (−1.08 + 1.08i)19-s + (1.37 + 1.37i)21-s + (0.772 − 0.772i)23-s + 0.199·27-s + (0.679 + 0.679i)29-s − 1.23i·31-s + (−0.905 + 0.905i)33-s − 0.180i·37-s + 1.57i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.812 - 0.583i$
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: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1548843948\)
\(L(\frac12)\) \(\approx\) \(0.1548843948\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.35T + 3T^{2} \)
7 \( 1 + (2.66 + 2.66i)T + 7iT^{2} \)
11 \( 1 + (-2.20 + 2.20i)T - 11iT^{2} \)
13 \( 1 + 4.16iT - 13T^{2} \)
17 \( 1 + (1.69 + 1.69i)T + 17iT^{2} \)
19 \( 1 + (4.74 - 4.74i)T - 19iT^{2} \)
23 \( 1 + (-3.70 + 3.70i)T - 23iT^{2} \)
29 \( 1 + (-3.65 - 3.65i)T + 29iT^{2} \)
31 \( 1 + 6.90iT - 31T^{2} \)
37 \( 1 + 1.10iT - 37T^{2} \)
41 \( 1 + 9.85iT - 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + (3.90 - 3.90i)T - 47iT^{2} \)
53 \( 1 + 6.19T + 53T^{2} \)
59 \( 1 + (3.42 + 3.42i)T + 59iT^{2} \)
61 \( 1 + (4.57 - 4.57i)T - 61iT^{2} \)
67 \( 1 - 6.37iT - 67T^{2} \)
71 \( 1 - 1.03T + 71T^{2} \)
73 \( 1 + (4.70 + 4.70i)T + 73iT^{2} \)
79 \( 1 + 2.54T + 79T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 + 1.77T + 89T^{2} \)
97 \( 1 + (1.16 + 1.16i)T + 97iT^{2} \)
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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

−9.029168830119732216231143914761, −8.048430083694529867994610871146, −7.04600029884221805350590991472, −6.31564072666969987810359822403, −5.93297731448263270160507700420, −4.80893553420891464974443280101, −3.93756043398100421834845780048, −2.94725447143285613580748989930, −1.01722456124191365546097857271, −0.087950168534269794961537814875, 1.68548294783599849848704380882, 2.94381401482238991601649831626, 4.31277061357395281378501599372, 4.97058982285656633828044253713, 5.99376855468535572974465916884, 6.64674287656647633694477319966, 6.91241218810877545073936849239, 8.558012427309920384743735224935, 9.169655554904686641046027010936, 9.844419076199860719026845683893

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