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Properties

Label 2-3344-209.87-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.420604448\)
\(L(\frac12)\) \(\approx\) \(1.420604448\)
\(L(1)\) 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

−8.874341981775792219496477495407, −7.71516293825422613506065632623, −7.41051504788649834913516470538, −6.90667618780919470374760878257, −6.04861242576437642792295666307, −4.94886598258231055810359141504, −4.08827741991653298849223850530, −3.02630257575934170294281118828, −2.43087150377381718953163530075, −1.29705064777516419515067770688, 0.928469578290736467609791843876, 2.34450152447720276718277655995, 3.65528509254898106820151039017, 3.95794247589038623986861427808, 4.69368519194251569367425411100, 5.75309273280014886013899050843, 6.38373671471685347373791308366, 7.47928268608528329477925247333, 8.383953509592628954854832660643, 8.960534680067239533974234021979

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