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Properties

Label 2-40e2-5.4-c1-0-26
Degree $2$
Conductor $1600$
Sign $-0.447 + 0.894i$
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  − 2i·3-s − 2i·7-s − 9-s + 4·11-s − 6i·13-s + 2i·17-s + 8·19-s − 4·21-s + 6i·23-s − 4i·27-s − 2·29-s + 4·31-s − 8i·33-s − 2i·37-s − 12·39-s + ⋯
L(s)  = 1  − 1.15i·3-s − 0.755i·7-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s + 0.485i·17-s + 1.83·19-s − 0.872·21-s + 1.25i·23-s − 0.769i·27-s − 0.371·29-s + 0.718·31-s − 1.39i·33-s − 0.328i·37-s − 1.92·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.904673010\)
\(L(\frac12)\) \(\approx\) \(1.904673010\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 6iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.117952306469587796120860827362, −8.024469776857550326543995507809, −7.53369881751463538510417247131, −6.92401728951112181007223304880, −6.01351032673423886002401589310, −5.21358882191819353482594271906, −3.86411930201301308374933655949, −3.10586961599280682809307628668, −1.56454295220466950950184753493, −0.850242754109977132122766858480, 1.50774740192706163630300199086, 2.88629165733295320274301106504, 3.89658893442426124530836401212, 4.60215591635517243512643127700, 5.36373816776797327755562939933, 6.47903964811604642931322440319, 7.08236231388169379700618667054, 8.384984644918352068433634988480, 9.256500221090967750008285562604, 9.403920090483631738283990942456

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