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Properties

Label 2-40e2-80.27-c1-0-25
Degree $2$
Conductor $1600$
Sign $0.0662 + 0.997i$
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  + 0.207·3-s + (1.32 + 1.32i)7-s − 2.95·9-s + (2.39 − 2.39i)11-s − 4.20i·13-s + (−3.29 − 3.29i)17-s + (0.838 − 0.838i)19-s + (0.274 + 0.274i)21-s + (−2.67 + 2.67i)23-s − 1.23·27-s + (−2.55 − 2.55i)29-s − 6.23i·31-s + (0.496 − 0.496i)33-s + 4.29i·37-s − 0.871i·39-s + ⋯
L(s)  = 1  + 0.119·3-s + (0.499 + 0.499i)7-s − 0.985·9-s + (0.721 − 0.721i)11-s − 1.16i·13-s + (−0.798 − 0.798i)17-s + (0.192 − 0.192i)19-s + (0.0598 + 0.0598i)21-s + (−0.557 + 0.557i)23-s − 0.237·27-s + (−0.474 − 0.474i)29-s − 1.12i·31-s + (0.0864 − 0.0864i)33-s + 0.706i·37-s − 0.139i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0662 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0662 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.0662 + 0.997i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.0662 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.364439242\)
\(L(\frac12)\) \(\approx\) \(1.364439242\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.207T + 3T^{2} \)
7 \( 1 + (-1.32 - 1.32i)T + 7iT^{2} \)
11 \( 1 + (-2.39 + 2.39i)T - 11iT^{2} \)
13 \( 1 + 4.20iT - 13T^{2} \)
17 \( 1 + (3.29 + 3.29i)T + 17iT^{2} \)
19 \( 1 + (-0.838 + 0.838i)T - 19iT^{2} \)
23 \( 1 + (2.67 - 2.67i)T - 23iT^{2} \)
29 \( 1 + (2.55 + 2.55i)T + 29iT^{2} \)
31 \( 1 + 6.23iT - 31T^{2} \)
37 \( 1 - 4.29iT - 37T^{2} \)
41 \( 1 + 7.06iT - 41T^{2} \)
43 \( 1 - 9.43iT - 43T^{2} \)
47 \( 1 + (-8.31 + 8.31i)T - 47iT^{2} \)
53 \( 1 - 7.66T + 53T^{2} \)
59 \( 1 + (7.07 + 7.07i)T + 59iT^{2} \)
61 \( 1 + (-8.74 + 8.74i)T - 61iT^{2} \)
67 \( 1 - 12.5iT - 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (4.79 + 4.79i)T + 73iT^{2} \)
79 \( 1 + 8.69T + 79T^{2} \)
83 \( 1 - 4.14T + 83T^{2} \)
89 \( 1 - 0.548T + 89T^{2} \)
97 \( 1 + (12.2 + 12.2i)T + 97iT^{2} \)
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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

−9.107488655671203529919620287969, −8.423205484567145393235798179852, −7.84829106651614020518266330709, −6.76742163734564957172803107287, −5.73752721862542950818896885594, −5.36490029918907741334802903931, −4.07677252448066074016672347030, −3.07839742395142952462450938380, −2.17911660583967419220318879739, −0.51915348659164094237828825302, 1.46839308687103936794508522125, 2.45464001562851312494838816147, 3.88727966016540900452205728692, 4.39863955196233611509755351450, 5.52333074839407517387966433671, 6.48912424969512261923542656581, 7.12862952159895486419017371519, 8.085522491968245320770910610672, 8.896786182857784488915939823067, 9.354734140680814065676090064079

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