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Properties

Label 2-1760-440.219-c1-0-51
Degree $2$
Conductor $1760$
Sign $0.292 + 0.956i$
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 − 5.22i·7-s + 3·9-s + 3.31·11-s − 3.56i·13-s − 4.55·17-s − 5.00·25-s − 8.94i·31-s + 11.6·35-s − 9.96·43-s + 6.70i·45-s − 20.2·49-s + 7.41i·55-s + 4·59-s − 15.6i·63-s + ⋯
L(s)  = 1  + 0.999i·5-s − 1.97i·7-s + 9-s + 1.00·11-s − 0.989i·13-s − 1.10·17-s − 1.00·25-s − 1.60i·31-s + 1.97·35-s − 1.51·43-s + 0.999i·45-s − 2.89·49-s + 0.999i·55-s + 0.520·59-s − 1.97i·63-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1760\)    =    \(2^{5} \cdot 5 \cdot 11\)
Sign: $0.292 + 0.956i$
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: \((2,\ 1760,\ (\ :1/2),\ 0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.686200637\)
\(L(\frac12)\) \(\approx\) \(1.686200637\)
\(L(\frac{3}{2})\) 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 \( 1 + 5.22iT - 7T^{2} \)
13 \( 1 + 3.56iT - 13T^{2} \)
17 \( 1 + 4.55T + 17T^{2} \)
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 \( 1 + 9.96T + 43T^{2} \)
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 \( 1 - 17.0T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
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

−9.445547266325023790417581952383, −8.030347446742556860086851920059, −7.48361851604946341195493909597, −6.74239265097759169873641023132, −6.34925320247788105555025181798, −4.78670062312779101918002803795, −3.96070241436589144071922054181, −3.46269347026103215323164127748, −1.94741683097878626597325397726, −0.65416509278143533865066245237, 1.50456906999660470192152643876, 2.20538633294883531618676592062, 3.69496953631199221302591573331, 4.67750584792565812344684798222, 5.23344372773756174191077280579, 6.36174490742613981299028037543, 6.83065092175873683048664192705, 8.209191876502090397529216815677, 8.854058307420262072611344536645, 9.213814396164932913874113888061

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