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Properties

Label 2-40e2-80.69-c1-0-25
Degree $2$
Conductor $1600$
Sign $0.323 + 0.946i$
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 − 2.73·7-s + 0.755i·9-s + (−4.12 − 4.12i)11-s + (1.37 + 1.37i)13-s − 4.94i·17-s + (−0.292 + 0.292i)19-s + (−3.74 − 3.74i)21-s + 1.64·23-s + (3.07 − 3.07i)27-s + (5.67 − 5.67i)29-s − 3.95·31-s − 11.2i·33-s + (−2.48 + 2.48i)37-s + 3.77i·39-s + ⋯
L(s)  = 1  + (0.791 + 0.791i)3-s − 1.03·7-s + 0.251i·9-s + (−1.24 − 1.24i)11-s + (0.382 + 0.382i)13-s − 1.20i·17-s + (−0.0671 + 0.0671i)19-s + (−0.817 − 0.817i)21-s + 0.343·23-s + (0.591 − 0.591i)27-s + (1.05 − 1.05i)29-s − 0.710·31-s − 1.96i·33-s + (−0.408 + 0.408i)37-s + 0.605i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.323 + 0.946i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.295214119\)
\(L(\frac12)\) \(\approx\) \(1.295214119\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + (4.12 + 4.12i)T + 11iT^{2} \)
13 \( 1 + (-1.37 - 1.37i)T + 13iT^{2} \)
17 \( 1 + 4.94iT - 17T^{2} \)
19 \( 1 + (0.292 - 0.292i)T - 19iT^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
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 \( 1 - 5.19iT - 47T^{2} \)
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 \( 1 - 4.36T + 73T^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 + (0.516 + 0.516i)T + 83iT^{2} \)
89 \( 1 - 6.42iT - 89T^{2} \)
97 \( 1 + 9.44iT - 97T^{2} \)
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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

−9.193432143047039880569883799094, −8.679318950782195169299657599194, −7.84996875978797348765879141681, −6.82787656756932022156954742978, −5.96220213665295167145733005717, −5.06205415001027606042977910743, −3.96103832176100197114228230394, −3.15607167254144245067836352174, −2.59654668658684817557385843124, −0.43772812438211909077303498246, 1.51460161948799117000607581391, 2.58693205173848645007121191225, 3.27158182371001068044387799377, 4.52411223994201357717135435813, 5.56798333055441308500175525126, 6.56137052807241192069578609280, 7.26295266821455625118118237813, 7.926694780942094606829191191610, 8.644747476457935565334336199071, 9.465685662464753525932272788956

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