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Properties

Label 2-40e2-16.13-c1-0-18
Degree $2$
Conductor $1600$
Sign $0.382 + 0.923i$
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 − i)3-s − i·9-s + (3 − 3i)11-s + (3 + 3i)13-s + 4·17-s + (−1 − i)19-s + 8i·23-s + (−4 + 4i)27-s + (−3 − 3i)29-s − 6·33-s + (3 − 3i)37-s − 6i·39-s + (3 − 3i)43-s − 2·47-s + 7·49-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s − 0.333i·9-s + (0.904 − 0.904i)11-s + (0.832 + 0.832i)13-s + 0.970·17-s + (−0.229 − 0.229i)19-s + 1.66i·23-s + (−0.769 + 0.769i)27-s + (−0.557 − 0.557i)29-s − 1.04·33-s + (0.493 − 0.493i)37-s − 0.960i·39-s + (0.457 − 0.457i)43-s − 0.291·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.382 + 0.923i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.495813695\)
\(L(\frac12)\) \(\approx\) \(1.495813695\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (-3 + 3i)T - 11iT^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + (1 + i)T + 19iT^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-3 + 3i)T - 43iT^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (-9 + 9i)T - 53iT^{2} \)
59 \( 1 + (-9 + 9i)T - 59iT^{2} \)
61 \( 1 + (5 + 5i)T + 61iT^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (9 + 9i)T + 83iT^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.218389907483313302817842802629, −8.542019788782969259470722357142, −7.49209862717289665177562414446, −6.78121466765948119591514506054, −5.97919179546714795891094934094, −5.51278277207434065107806723603, −3.98189993015752885571722268460, −3.44620474159706869408090868779, −1.76296657973499314889894800984, −0.78693737983005232923352362527, 1.13508494439198558991427077677, 2.56127311830190730740249320736, 3.85953434011924264483549117304, 4.51333780310260280007689941975, 5.49392581475311240114600842940, 6.14098217292904939195342161985, 7.13308311053660077800456212812, 7.998065423440694439071413149315, 8.812879179883277481024097487469, 9.702758765679975250072566101202

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