Properties

Label 2-40e2-16.5-c1-0-32
Degree $2$
Conductor $1600$
Sign $-0.701 + 0.712i$
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  + (1.37 − 1.37i)3-s − 2.73i·7-s − 0.755i·9-s + (−4.12 − 4.12i)11-s + (1.37 − 1.37i)13-s + 4.94·17-s + (0.292 − 0.292i)19-s + (−3.74 − 3.74i)21-s − 1.64i·23-s + (3.07 + 3.07i)27-s + (−5.67 + 5.67i)29-s − 3.95·31-s − 11.2·33-s + (−2.48 − 2.48i)37-s − 3.77i·39-s + ⋯
L(s)  = 1  + (0.791 − 0.791i)3-s − 1.03i·7-s − 0.251i·9-s + (−1.24 − 1.24i)11-s + (0.382 − 0.382i)13-s + 1.20·17-s + (0.0671 − 0.0671i)19-s + (−0.817 − 0.817i)21-s − 0.343i·23-s + (0.591 + 0.591i)27-s + (−1.05 + 1.05i)29-s − 0.710·31-s − 1.96·33-s + (−0.408 − 0.408i)37-s − 0.605i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.811943389\)
\(L(\frac12)\) \(\approx\) \(1.811943389\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 1.37i)T - 3iT^{2} \)
7 \( 1 + 2.73iT - 7T^{2} \)
11 \( 1 + (4.12 + 4.12i)T + 11iT^{2} \)
13 \( 1 + (-1.37 + 1.37i)T - 13iT^{2} \)
17 \( 1 - 4.94T + 17T^{2} \)
19 \( 1 + (-0.292 + 0.292i)T - 19iT^{2} \)
23 \( 1 + 1.64iT - 23T^{2} \)
29 \( 1 + (5.67 - 5.67i)T - 29iT^{2} \)
31 \( 1 + 3.95T + 31T^{2} \)
37 \( 1 + (2.48 + 2.48i)T + 37iT^{2} \)
41 \( 1 + 8.40iT - 41T^{2} \)
43 \( 1 + (3.22 + 3.22i)T + 43iT^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 + (7.20 + 7.20i)T + 53iT^{2} \)
59 \( 1 + (-6.41 - 6.41i)T + 59iT^{2} \)
61 \( 1 + (3.82 - 3.82i)T - 61iT^{2} \)
67 \( 1 + (-5.76 + 5.76i)T - 67iT^{2} \)
71 \( 1 - 7.92iT - 71T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (0.516 - 0.516i)T - 83iT^{2} \)
89 \( 1 + 6.42iT - 89T^{2} \)
97 \( 1 - 9.44T + 97T^{2} \)
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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

−8.774658162302011273768525714708, −8.260996435576978488159911029361, −7.50930098435136281337868329274, −7.09087331510515258632775295094, −5.79485869294029414224004415972, −5.13450929486102504845117088248, −3.60199078846955758468073132236, −3.13148653673971385050905992988, −1.84975858055473398903761355323, −0.61190073927350273189478545732, 1.88076600260289549515283974264, 2.82903032381019371007252456919, 3.66551719098354202107987022735, 4.73819099327621521741891198353, 5.43056265744847248866177644897, 6.40414097630944250471372517000, 7.66241956945702132163275919422, 8.103923479503690310856034203302, 9.087589673036037892221451285263, 9.694274846095777872908738773041

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