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Properties

Label 2-40e2-40.29-c1-0-26
Degree $2$
Conductor $1600$
Sign $-0.948 - 0.316i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 6i·11-s + 6i·17-s − 2i·19-s + 4·27-s + 12i·33-s − 6·41-s − 10·43-s + 7·49-s − 12i·51-s + 4i·57-s + 6i·59-s − 14·67-s + 2i·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.333·9-s − 1.80i·11-s + 1.45i·17-s − 0.458i·19-s + 0.769·27-s + 2.08i·33-s − 0.937·41-s − 1.52·43-s + 49-s − 1.68i·51-s + 0.529i·57-s + 0.781i·59-s − 1.71·67-s + 0.234i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.632998009671044337676015389991, −8.493264023253021968128737306529, −7.19053170722770689998093942716, −6.21806256559437420914353141876, −5.86527169376354659790671520497, −5.02968443177532664818177209995, −3.90871129395749100222680498506, −2.93801831066259447993989928945, −1.29213611856867358002892438412, 0, 1.60047304650876806231188647670, 2.86502747756074582961215094783, 4.29779990415778891390719514373, 4.98450710694390351225680959151, 5.64006864436831248426242047187, 6.78432884768124675850943487843, 7.12083300677956764268778798250, 8.167837006198653615517623175890, 9.269512082871218195265542415710

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