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Properties

Label 2-40e2-40.3-c1-0-26
Degree $2$
Conductor $1600$
Sign $-0.945 + 0.326i$
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  + (−1.87 − 1.87i)3-s + (−2.44 − 2.44i)7-s + 4i·9-s + 4.58·11-s + (3.74 − 3.74i)13-s + (−1.22 + 1.22i)17-s − 4.58i·19-s + 9.16i·21-s + (4.89 − 4.89i)23-s + (1.87 − 1.87i)27-s + 9.16·29-s − 4i·31-s + (−8.57 − 8.57i)33-s + (3.74 + 3.74i)37-s − 14·39-s + ⋯
L(s)  = 1  + (−1.08 − 1.08i)3-s + (−0.925 − 0.925i)7-s + 1.33i·9-s + 1.38·11-s + (1.03 − 1.03i)13-s + (−0.297 + 0.297i)17-s − 1.05i·19-s + 1.99i·21-s + (1.02 − 1.02i)23-s + (0.360 − 0.360i)27-s + 1.70·29-s − 0.718i·31-s + (−1.49 − 1.49i)33-s + (0.615 + 0.615i)37-s − 2.24·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.945 + 0.326i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.011314424\)
\(L(\frac12)\) \(\approx\) \(1.011314424\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.87 + 1.87i)T + 3iT^{2} \)
7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 - 4.58T + 11T^{2} \)
13 \( 1 + (-3.74 + 3.74i)T - 13iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - 17iT^{2} \)
19 \( 1 + 4.58iT - 19T^{2} \)
23 \( 1 + (-4.89 + 4.89i)T - 23iT^{2} \)
29 \( 1 - 9.16T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + (3.74 + 3.74i)T + 43iT^{2} \)
47 \( 1 + (-2.44 - 2.44i)T + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 9.16iT - 61T^{2} \)
67 \( 1 + (1.87 - 1.87i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (6.12 + 6.12i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-5.61 - 5.61i)T + 83iT^{2} \)
89 \( 1 - 15iT - 89T^{2} \)
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

−8.961672327292031816738800381653, −8.170443648108537575869173811161, −7.03482552979911574726482157053, −6.56884803544780683120501500974, −6.23741090044988031068190316722, −5.03580141671912819064258495228, −3.98703623602430588059835672081, −2.92828980737376242244580247117, −1.23146055851410864603688768111, −0.55810314764018655559333690097, 1.40769714411135803241506689701, 3.17525031970325778358726416539, 3.95579125768195886836809021208, 4.79137274162124083140202960120, 5.81247252424720494202878340571, 6.30269441778229324918545540925, 6.95627105207344940092792963598, 8.644433940855180634734688784989, 9.076235643712610689968878408277, 9.772071590246203797870705076889

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