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Properties

Label 2-40e2-16.13-c1-0-19
Degree $2$
Conductor $1600$
Sign $0.453 - 0.891i$
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  + (1.99 + 1.99i)3-s − 1.09i·7-s + 4.93i·9-s + (2.33 − 2.33i)11-s + (1.80 + 1.80i)13-s + 4.93·17-s + (2.03 + 2.03i)19-s + (2.17 − 2.17i)21-s − 1.45i·23-s + (−3.84 + 3.84i)27-s + (0.707 + 0.707i)29-s − 10.1·31-s + 9.28·33-s + (4.35 − 4.35i)37-s + 7.20i·39-s + ⋯
L(s)  = 1  + (1.14 + 1.14i)3-s − 0.412i·7-s + 1.64i·9-s + (0.703 − 0.703i)11-s + (0.501 + 0.501i)13-s + 1.19·17-s + (0.467 + 0.467i)19-s + (0.473 − 0.473i)21-s − 0.303i·23-s + (−0.740 + 0.740i)27-s + (0.131 + 0.131i)29-s − 1.81·31-s + 1.61·33-s + (0.715 − 0.715i)37-s + 1.15i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.805530291\)
\(L(\frac12)\) \(\approx\) \(2.805530291\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 + (-2.33 + 2.33i)T - 11iT^{2} \)
13 \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \)
17 \( 1 - 4.93T + 17T^{2} \)
19 \( 1 + (-2.03 - 2.03i)T + 19iT^{2} \)
23 \( 1 + 1.45iT - 23T^{2} \)
29 \( 1 + (-0.707 - 0.707i)T + 29iT^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + (-4.35 + 4.35i)T - 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (2.22 - 2.22i)T - 43iT^{2} \)
47 \( 1 - 2.09T + 47T^{2} \)
53 \( 1 + (0.215 - 0.215i)T - 53iT^{2} \)
59 \( 1 + (1.16 - 1.16i)T - 59iT^{2} \)
61 \( 1 + (3.46 + 3.46i)T + 61iT^{2} \)
67 \( 1 + (5.04 + 5.04i)T + 67iT^{2} \)
71 \( 1 - 6.40iT - 71T^{2} \)
73 \( 1 - 5.24iT - 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 + (5.67 + 5.67i)T + 83iT^{2} \)
89 \( 1 - 6.87iT - 89T^{2} \)
97 \( 1 + 3.77T + 97T^{2} \)
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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

−9.434566661314781826524548452712, −8.931958877589818847662686606577, −8.119267432116915906103064875925, −7.43096352990879235889543986885, −6.23614325799982759250820057514, −5.30319121508488276559143385421, −4.16732975658350618458424294737, −3.66807552410595607037960282925, −2.88880479782383089479143419508, −1.40477190237938111835545520982, 1.13885118340022406265081448761, 2.06473753549833158423968378771, 3.09066133654395080120289193609, 3.85746532425475331751077584727, 5.30825551074671477580019289427, 6.16670439577228257469402114130, 7.25353162872974992377349924802, 7.50144496678463738839280755434, 8.489702385004316986260270784924, 9.075756299481641394805210266494

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