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Properties

Label 2-1040-13.12-c1-0-25
Degree $2$
Conductor $1040$
Sign $0.349 + 0.936i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + 2.26·3-s − i·5-s − 1.11i·7-s + 2.11·9-s − 5.37i·11-s + (−3.37 + 1.26i)13-s − 2.26i·15-s − 7.90i·19-s − 2.52i·21-s + 6.49·23-s − 25-s − 2·27-s + 3.63·29-s + 3.14i·31-s − 12.1i·33-s + ⋯
L(s)  = 1  + 1.30·3-s − 0.447i·5-s − 0.421i·7-s + 0.705·9-s − 1.62i·11-s + (−0.936 + 0.349i)13-s − 0.583i·15-s − 1.81i·19-s − 0.550i·21-s + 1.35·23-s − 0.200·25-s − 0.384·27-s + 0.675·29-s + 0.565i·31-s − 2.11i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.293027421\)
\(L(\frac12)\) \(\approx\) \(2.293027421\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (3.37 - 1.26i)T \)
good3 \( 1 - 2.26T + 3T^{2} \)
7 \( 1 + 1.11iT - 7T^{2} \)
11 \( 1 + 5.37iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 7.90iT - 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 - 3.63T + 29T^{2} \)
31 \( 1 - 3.14iT - 31T^{2} \)
37 \( 1 - 7.40iT - 37T^{2} \)
41 \( 1 + 4.75iT - 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 - 6.16iT - 47T^{2} \)
53 \( 1 - 0.292T + 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 + 5.34T + 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 + 3.43iT - 71T^{2} \)
73 \( 1 - 4.59iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 7.40iT - 83T^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−9.306747459491350378294347767499, −9.017052950525213403665504918624, −8.282730337693134037581288513737, −7.42032360163221821551345554446, −6.59600358645890238999413439034, −5.27394671859625069334500011943, −4.38998616756423445305712676529, −3.19253837355735307458783699750, −2.59751823498744246156671535841, −0.898801915960010559854677642732, 1.93120336970853949022758509559, 2.64461066942963482326557218119, 3.66578773686151426631565460671, 4.70704040593572741343587210945, 5.81514872272525066745888227238, 7.10253466047483361015329330512, 7.59197311056111513998269121138, 8.395274569174891370428890809686, 9.360195797335540547941286110031, 9.848838394885805790668626878664

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