Properties

Label 2-40e2-80.3-c1-0-26
Degree $2$
Conductor $1600$
Sign $0.630 + 0.776i$
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  + 1.54·3-s + (1.17 − 1.17i)7-s − 0.610·9-s + (−2.04 − 2.04i)11-s − 2.14i·13-s + (2.07 − 2.07i)17-s + (4.47 + 4.47i)19-s + (1.81 − 1.81i)21-s + (4.86 + 4.86i)23-s − 5.58·27-s + (5.51 − 5.51i)29-s − 5.72i·31-s + (−3.16 − 3.16i)33-s − 11.0i·37-s − 3.31i·39-s + ⋯
L(s)  = 1  + 0.892·3-s + (0.443 − 0.443i)7-s − 0.203·9-s + (−0.616 − 0.616i)11-s − 0.594i·13-s + (0.502 − 0.502i)17-s + (1.02 + 1.02i)19-s + (0.396 − 0.396i)21-s + (1.01 + 1.01i)23-s − 1.07·27-s + (1.02 − 1.02i)29-s − 1.02i·31-s + (−0.550 − 0.550i)33-s − 1.81i·37-s − 0.530i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.630 + 0.776i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.630 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.279858400\)
\(L(\frac12)\) \(\approx\) \(2.279858400\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.54T + 3T^{2} \)
7 \( 1 + (-1.17 + 1.17i)T - 7iT^{2} \)
11 \( 1 + (2.04 + 2.04i)T + 11iT^{2} \)
13 \( 1 + 2.14iT - 13T^{2} \)
17 \( 1 + (-2.07 + 2.07i)T - 17iT^{2} \)
19 \( 1 + (-4.47 - 4.47i)T + 19iT^{2} \)
23 \( 1 + (-4.86 - 4.86i)T + 23iT^{2} \)
29 \( 1 + (-5.51 + 5.51i)T - 29iT^{2} \)
31 \( 1 + 5.72iT - 31T^{2} \)
37 \( 1 + 11.0iT - 37T^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 - 0.251iT - 43T^{2} \)
47 \( 1 + (-0.119 - 0.119i)T + 47iT^{2} \)
53 \( 1 - 2.69T + 53T^{2} \)
59 \( 1 + (-1.24 + 1.24i)T - 59iT^{2} \)
61 \( 1 + (2.48 + 2.48i)T + 61iT^{2} \)
67 \( 1 - 9.23iT - 67T^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (7.85 - 7.85i)T - 73iT^{2} \)
79 \( 1 - 4.86T + 79T^{2} \)
83 \( 1 + 4.94T + 83T^{2} \)
89 \( 1 - 3.63T + 89T^{2} \)
97 \( 1 + (9.89 - 9.89i)T - 97iT^{2} \)
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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

−9.264718449163028708944011638744, −8.384709520325906026527314747964, −7.73976313313217369884720551324, −7.32152416580637650730266056564, −5.75441390489605761334980219971, −5.38223806830589462355219043462, −3.98286271486852243321235775348, −3.22501435657846594552084897346, −2.35685137412104699482381057327, −0.848646988715117127273962968312, 1.43742945793807177379903834831, 2.69141914187063312324347902766, 3.19169356974373412337600430988, 4.70568269714895085833597743237, 5.12834454373415385249275937948, 6.45057424631483194069087764265, 7.21081536354250993094965145847, 8.172812089988335776069528053088, 8.620109684661368157857393052791, 9.364819249284244867997545479336

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