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Properties

Label 2-40e2-20.3-c1-0-30
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  + (2.22 − 2.22i)3-s + (2 + 2i)7-s − 6.89i·9-s − 3.44i·11-s + (−4.44 − 4.44i)13-s + (−0.775 + 0.775i)17-s − 2.55·19-s + 8.89·21-s + (0.449 − 0.449i)23-s + (−8.67 − 8.67i)27-s + 2.89i·29-s + 2.89i·31-s + (−7.67 − 7.67i)33-s + (6 − 6i)37-s − 19.7·39-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)3-s + (0.755 + 0.755i)7-s − 2.29i·9-s − 1.04i·11-s + (−1.23 − 1.23i)13-s + (−0.188 + 0.188i)17-s − 0.585·19-s + 1.94·21-s + (0.0937 − 0.0937i)23-s + (−1.66 − 1.66i)27-s + 0.538i·29-s + 0.520i·31-s + (−1.33 − 1.33i)33-s + (0.986 − 0.986i)37-s − 3.17·39-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

\(L(1)\) \(\approx\) \(2.564225112\)
\(L(\frac12)\) \(\approx\) \(2.564225112\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 2.22i)T - 3iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 3.44iT - 11T^{2} \)
13 \( 1 + (4.44 + 4.44i)T + 13iT^{2} \)
17 \( 1 + (0.775 - 0.775i)T - 17iT^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 + (-0.449 + 0.449i)T - 23iT^{2} \)
29 \( 1 - 2.89iT - 29T^{2} \)
31 \( 1 - 2.89iT - 31T^{2} \)
37 \( 1 + (-6 + 6i)T - 37iT^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + (-2.89 + 2.89i)T - 43iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 47iT^{2} \)
53 \( 1 + (-6.44 - 6.44i)T + 53iT^{2} \)
59 \( 1 - 7.79T + 59T^{2} \)
61 \( 1 - 0.898T + 61T^{2} \)
67 \( 1 + (-4.22 - 4.22i)T + 67iT^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (-5.67 - 5.67i)T + 73iT^{2} \)
79 \( 1 - 0.898T + 79T^{2} \)
83 \( 1 + (-4.67 + 4.67i)T - 83iT^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
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

−8.712211344042555802922721743274, −8.419776852575858193756489827211, −7.69718094036766706976030416943, −7.01363668461690278251171627154, −5.98067923478083920555253865189, −5.18418106029440650458433777534, −3.73558470578260884849002753413, −2.69923005278298236605425852143, −2.19486171536224676955764049623, −0.825913369565941238279379737066, 1.92612035014762841065554449312, 2.65795503160057698719055425589, 4.04573213808522257409008805859, 4.45151598215290668023308600529, 5.00150591070877674288942659308, 6.66265940986880257596001306400, 7.61081784946898953371092568214, 8.037626230940400598348296593047, 9.103063242155229654656868541433, 9.649230171234792645667385755969

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