Properties

Label 2-40e2-40.29-c1-0-10
Degree $2$
Conductor $1600$
Sign $0.316 - 0.948i$
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  + 1.44·3-s − 0.898·9-s + 0.550i·11-s + 7.89i·17-s + 8.34i·19-s − 5.65·27-s + 0.797i·33-s + 12.7·41-s + 10·43-s + 7·49-s + 11.4i·51-s + 12.1i·57-s + 6i·59-s − 14.3·67-s − 13.6i·73-s + ⋯
L(s)  = 1  + 0.836·3-s − 0.299·9-s + 0.165i·11-s + 1.91i·17-s + 1.91i·19-s − 1.08·27-s + 0.138i·33-s + 1.99·41-s + 1.52·43-s + 49-s + 1.60i·51-s + 1.60i·57-s + 0.781i·59-s − 1.75·67-s − 1.60i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.907299712\)
\(L(\frac12)\) \(\approx\) \(1.907299712\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44T + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 0.550iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.89iT - 17T^{2} \)
19 \( 1 - 8.34iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 12.7T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
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

−9.394289226250301613254076990907, −8.742384915950481917328003795252, −7.977203159571139344409820308906, −7.52343773232282066242295530663, −6.07015332757248056055328264271, −5.78898511509479424494239835075, −4.22395190518720324456351522912, −3.66744709256206539299879682834, −2.54169160458573523186853681394, −1.53663954806448441323743265768, 0.66375882600806276559657655504, 2.53551104521227321984338282045, 2.85186641827609316091127985000, 4.15345079921294696344276114530, 5.05066732546330666237271840380, 5.97084569661213222854921567087, 7.15317558683897499190492050334, 7.54626438475021572319702459089, 8.676549636472985802397198225052, 9.158545144832157480576799647968

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