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Properties

Label 2-1760-440.219-c1-0-39
Degree $2$
Conductor $1760$
Sign $0.956 + 0.292i$
Analytic cond. $14.0536$
Root an. cond. $3.74882$
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  + 2.23i·5-s − 0.856i·7-s + 3·9-s − 3.31·11-s − 6.26i·13-s + 6.87·17-s − 5.00·25-s − 8.94i·31-s + 1.91·35-s + 8.52·43-s + 6.70i·45-s + 6.26·49-s − 7.41i·55-s + 4·59-s − 2.56i·63-s + ⋯
L(s)  = 1  + 0.999i·5-s − 0.323i·7-s + 9-s − 1.00·11-s − 1.73i·13-s + 1.66·17-s − 1.00·25-s − 1.60i·31-s + 0.323·35-s + 1.30·43-s + 0.999i·45-s + 0.895·49-s − 0.999i·55-s + 0.520·59-s − 0.323i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1760\)    =    \(2^{5} \cdot 5 \cdot 11\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(14.0536\)
Root analytic conductor: \(3.74882\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1760} (879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1760,\ (\ :1/2),\ 0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785406832\)
\(L(\frac12)\) \(\approx\) \(1.785406832\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
11 \( 1 + 3.31T \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + 0.856iT - 7T^{2} \)
13 \( 1 + 6.26iT - 13T^{2} \)
17 \( 1 - 6.87T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.94iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 8.52T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 - 0.261T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.596330269617482204635759755747, −8.084377620849351780985811170459, −7.67663460218969479154680973961, −7.13056082660555410424607557888, −5.90038892470565251894729134868, −5.40973591133311934701795843266, −4.11634842630608740373290757545, −3.25224391446947849719069425520, −2.41627868069008889547525042718, −0.808546003440423809980394154175, 1.14504567082049547791054917507, 2.11628223728650264840032888236, 3.53996608564832832692130425813, 4.51185833430750500405981398792, 5.13871028947801135481542839989, 6.01682336930457203116538102494, 7.11441722335080251334275723394, 7.73729879408603689940929588966, 8.641236227543284590546001286858, 9.316845946110165033433302509672

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