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Properties

Label 2-40e2-80.29-c1-0-31
Degree $2$
Conductor $1600$
Sign $-0.543 + 0.839i$
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  + (1.66 − 1.66i)3-s − 1.87·7-s − 2.56i·9-s + (3.29 − 3.29i)11-s + (−1.90 + 1.90i)13-s − 2.57i·17-s + (−5.76 − 5.76i)19-s + (−3.12 + 3.12i)21-s + 7.58·23-s + (0.728 + 0.728i)27-s + (−6.45 − 6.45i)29-s + 0.799·31-s − 10.9i·33-s + (−2.69 − 2.69i)37-s + 6.33i·39-s + ⋯
L(s)  = 1  + (0.962 − 0.962i)3-s − 0.708·7-s − 0.854i·9-s + (0.994 − 0.994i)11-s + (−0.527 + 0.527i)13-s − 0.623i·17-s + (−1.32 − 1.32i)19-s + (−0.681 + 0.681i)21-s + 1.58·23-s + (0.140 + 0.140i)27-s + (−1.19 − 1.19i)29-s + 0.143·31-s − 1.91i·33-s + (−0.443 − 0.443i)37-s + 1.01i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.916189923\)
\(L(\frac12)\) \(\approx\) \(1.916189923\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 1.66i)T - 3iT^{2} \)
7 \( 1 + 1.87T + 7T^{2} \)
11 \( 1 + (-3.29 + 3.29i)T - 11iT^{2} \)
13 \( 1 + (1.90 - 1.90i)T - 13iT^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 + (5.76 + 5.76i)T + 19iT^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + (6.45 + 6.45i)T + 29iT^{2} \)
31 \( 1 - 0.799T + 31T^{2} \)
37 \( 1 + (2.69 + 2.69i)T + 37iT^{2} \)
41 \( 1 + 0.946iT - 41T^{2} \)
43 \( 1 + (0.829 + 0.829i)T + 43iT^{2} \)
47 \( 1 - 1.52iT - 47T^{2} \)
53 \( 1 + (-6.97 - 6.97i)T + 53iT^{2} \)
59 \( 1 + (-6.84 + 6.84i)T - 59iT^{2} \)
61 \( 1 + (6.87 + 6.87i)T + 61iT^{2} \)
67 \( 1 + (-3.73 + 3.73i)T - 67iT^{2} \)
71 \( 1 + 9.34iT - 71T^{2} \)
73 \( 1 + 0.886T + 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + (0.989 - 0.989i)T - 83iT^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 - 7.16iT - 97T^{2} \)
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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

−9.145180144419305417404139039647, −8.447271553514187224254218581993, −7.44149875690956650982927475609, −6.80312490316614171071047620852, −6.27723042661026239224592734201, −4.94852822112451758619841734348, −3.78522449278479206424829656544, −2.88796724213446062529339126232, −2.07081951958392552020460874431, −0.63893469279187070033072201766, 1.73265013293495028997959826189, 2.95911220811958654174755962058, 3.75352842914309053024615724800, 4.38897402486838302924430452769, 5.46465210110228628173956985564, 6.59121756024564222056799042032, 7.26982606753888196841200557364, 8.438527731442252698737255820833, 8.896862474595105735159060123931, 9.740854119180526440926791425463

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