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Properties

Label 2-1760-440.219-c1-0-27
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

\(L(1)\) \(\approx\) \(2.032924464\)
\(L(\frac12)\) \(\approx\) \(2.032924464\)
\(L(\frac{3}{2})\) 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

−9.453852155413049246013204850929, −8.981932096336763891668861023633, −7.88255476742088956248274277261, −7.11342629533237057385303210786, −6.20899884660995773969755093142, −5.79242964465647830799481500966, −4.52276021343045094288858336585, −3.58918717266500860981810226850, −2.51609096073186161553530089130, −1.69353556393001229245365918610, 0.906276248942401460452629944313, 1.35316850333099003328013847451, 3.40415742776858760367402822902, 4.09795145286283222162227739443, 4.74339535010528851840783208945, 5.77923846941272284637732046850, 6.93186561085721978754779258903, 7.43416735126992707614803714812, 8.163901839655588862737337440340, 9.123042673341433469505380770965

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