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Properties

Label 2-40e2-20.3-c1-0-27
Degree $2$
Conductor $1600$
Sign $-0.437 + 0.899i$
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  + (0.224 − 0.224i)3-s + (−2 − 2i)7-s + 2.89i·9-s + 1.44i·11-s + (−0.449 − 0.449i)13-s + (3.22 − 3.22i)17-s − 7.44·19-s − 0.898·21-s + (4.44 − 4.44i)23-s + (1.32 + 1.32i)27-s − 6.89i·29-s − 6.89i·31-s + (0.325 + 0.325i)33-s + (−6 + 6i)37-s − 0.202·39-s + ⋯
L(s)  = 1  + (0.129 − 0.129i)3-s + (−0.755 − 0.755i)7-s + 0.966i·9-s + 0.437i·11-s + (−0.124 − 0.124i)13-s + (0.782 − 0.782i)17-s − 1.70·19-s − 0.196·21-s + (0.927 − 0.927i)23-s + (0.255 + 0.255i)27-s − 1.28i·29-s − 1.23i·31-s + (0.0567 + 0.0567i)33-s + (−0.986 + 0.986i)37-s − 0.0323·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.437 + 0.899i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9599774752\)
\(L(\frac12)\) \(\approx\) \(0.9599774752\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.224 + 0.224i)T - 3iT^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - 1.44iT - 11T^{2} \)
13 \( 1 + (0.449 + 0.449i)T + 13iT^{2} \)
17 \( 1 + (-3.22 + 3.22i)T - 17iT^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 + (-4.44 + 4.44i)T - 23iT^{2} \)
29 \( 1 + 6.89iT - 29T^{2} \)
31 \( 1 + 6.89iT - 31T^{2} \)
37 \( 1 + (6 - 6i)T - 37iT^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + (-6.89 + 6.89i)T - 43iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 47iT^{2} \)
53 \( 1 + (1.55 + 1.55i)T + 53iT^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 8.89T + 61T^{2} \)
67 \( 1 + (1.77 + 1.77i)T + 67iT^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (-1.67 - 1.67i)T + 73iT^{2} \)
79 \( 1 + 8.89T + 79T^{2} \)
83 \( 1 + (-2.67 + 2.67i)T - 83iT^{2} \)
89 \( 1 - 2.10iT - 89T^{2} \)
97 \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \)
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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

−9.188111758406297680641784541840, −8.259422701562536938985422031705, −7.52760594621624816918469258877, −6.84217947419010307176186984197, −6.00295425977502226278823830569, −4.85650930802647569518244978098, −4.17382426266118325683457497278, −2.97979173962494175270968556176, −2.04067391475442019996230096360, −0.36224083690272387276183243660, 1.44514838645531214487129193823, 2.95042058528802793501093295407, 3.50216696254177030269517610449, 4.64487282322130063525072692462, 5.83144375592378824907675887362, 6.27697025022738304739812772138, 7.17373812154456632106734311659, 8.253866238172006777487568077530, 9.174062663049260696154776015901, 9.249367127347916501247716196789

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