Properties

Label 2-3344-76.75-c1-0-60
Degree $2$
Conductor $3344$
Sign $0.827 + 0.561i$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
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  − 3.02·3-s + 2.92·5-s + 2.58i·7-s + 6.15·9-s + i·11-s − 1.48i·13-s − 8.84·15-s + 0.518·17-s + (2.44 − 3.60i)19-s − 7.83i·21-s + 0.564i·23-s + 3.54·25-s − 9.56·27-s − 7.78i·29-s + 3.27·31-s + ⋯
L(s)  = 1  − 1.74·3-s + 1.30·5-s + 0.978i·7-s + 2.05·9-s + 0.301i·11-s − 0.413i·13-s − 2.28·15-s + 0.125·17-s + (0.561 − 0.827i)19-s − 1.70i·21-s + 0.117i·23-s + 0.708·25-s − 1.84·27-s − 1.44i·29-s + 0.587·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(26.7019\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218211158\)
\(L(\frac12)\) \(\approx\) \(1.218211158\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - iT \)
19 \( 1 + (-2.44 + 3.60i)T \)
good3 \( 1 + 3.02T + 3T^{2} \)
5 \( 1 - 2.92T + 5T^{2} \)
7 \( 1 - 2.58iT - 7T^{2} \)
13 \( 1 + 1.48iT - 13T^{2} \)
17 \( 1 - 0.518T + 17T^{2} \)
23 \( 1 - 0.564iT - 23T^{2} \)
29 \( 1 + 7.78iT - 29T^{2} \)
31 \( 1 - 3.27T + 31T^{2} \)
37 \( 1 + 5.18iT - 37T^{2} \)
41 \( 1 + 1.05iT - 41T^{2} \)
43 \( 1 + 4.86iT - 43T^{2} \)
47 \( 1 + 13.1iT - 47T^{2} \)
53 \( 1 - 6.61iT - 53T^{2} \)
59 \( 1 + 1.16T + 59T^{2} \)
61 \( 1 - 4.69T + 61T^{2} \)
67 \( 1 + 1.34T + 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 - 1.60iT - 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 - 8.44iT - 97T^{2} \)
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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

−8.769928694807839099695190476472, −7.55811336264638614311838962642, −6.76139113232382423123290010369, −6.06201322377680828305704691756, −5.54645555556331081859226539060, −5.18319459875799137919403494158, −4.19398054395861405871543228968, −2.66440041749012974938145271005, −1.81029555324968343648508555843, −0.57541763111581775635202594870, 1.00396479163129936646689875389, 1.63228075221475393983826288387, 3.19838294829741584735352073615, 4.40533119863637568118359096435, 4.98497048158511454627646187295, 5.85930341341532698663966472354, 6.21292249542772131272811800949, 6.95718988625498591826663010195, 7.65040673469907591004109234437, 8.855865161169004175336003833441

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