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Properties

Label 2-3344-209.87-c0-0-0
Degree $2$
Conductor $3344$
Sign $0.977 - 0.211i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s − i·11-s + (−0.866 + 0.5i)13-s + (0.499 + 0.866i)15-s + (0.866 + 0.5i)17-s + i·19-s + (0.5 + 0.866i)23-s + 27-s + (0.866 − 0.5i)29-s + (−0.866 − 0.5i)33-s + 0.999i·39-s + (−0.866 − 0.5i)41-s + (−0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s − i·11-s + (−0.866 + 0.5i)13-s + (0.499 + 0.866i)15-s + (0.866 + 0.5i)17-s + i·19-s + (0.5 + 0.866i)23-s + 27-s + (0.866 − 0.5i)29-s + (−0.866 − 0.5i)33-s + 0.999i·39-s + (−0.866 − 0.5i)41-s + (−0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.977 - 0.211i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.375189871\)
\(L(\frac12)\) \(\approx\) \(1.375189871\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.517416827633512159712175444188, −8.011454873772454541763017586781, −7.34323259747414347110373591513, −6.85451754864455689302578853188, −5.97576308029341217039797616096, −5.11900789331847829046429585526, −3.87516959937932519979787925967, −3.20484880912482043521842449306, −2.36851645714647463068376422587, −1.28589456766136242926587552437, 0.890220590124699214964410603576, 2.49043327343045145903628848680, 3.26560848321608844321436234506, 4.34378319724158261628181340768, 4.76174040715872709046447996045, 5.35259365460966133336164992973, 6.81029200112516913106646705059, 7.26241485629536208578287550465, 8.335558241688771797563284551926, 8.718990585860828393543992585717

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