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Properties

Label 2-3344-209.75-c0-0-2
Degree $2$
Conductor $3344$
Sign $-0.935 + 0.352i$
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 − 1.53i)5-s + (−1.30 − 0.951i)7-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)11-s + (0.190 + 0.587i)17-s + (0.809 − 0.587i)19-s − 0.618·23-s + (−1.30 + 0.951i)25-s + (−0.809 + 2.48i)35-s − 0.618·43-s − 1.61·45-s + (0.5 − 0.363i)47-s + (0.500 + 1.53i)49-s + (−1.30 − 0.951i)55-s + (0.190 + 0.587i)61-s + ⋯
L(s)  = 1  + (−0.5 − 1.53i)5-s + (−1.30 − 0.951i)7-s + (0.309 − 0.951i)9-s + (0.809 − 0.587i)11-s + (0.190 + 0.587i)17-s + (0.809 − 0.587i)19-s − 0.618·23-s + (−1.30 + 0.951i)25-s + (−0.809 + 2.48i)35-s − 0.618·43-s − 1.61·45-s + (0.5 − 0.363i)47-s + (0.500 + 1.53i)49-s + (−1.30 − 0.951i)55-s + (0.190 + 0.587i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ -0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8673151481\)
\(L(\frac12)\) \(\approx\) \(0.8673151481\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + 0.618T + T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−8.637957690464519755423606929663, −7.75762392110588535472304968694, −6.93332292384684876681063197152, −6.29197738372275540442083208891, −5.47756382702534826830447552648, −4.35056068526401578134553940145, −3.85617831904522640393957980700, −3.21679502255569255766303647369, −1.32803341420726037658556661315, −0.55804812443068504689411330649, 1.95713476896989397590134376138, 2.86028911477654288620449905110, 3.42795533578020552661554301121, 4.37252513095700784006022707892, 5.56504516159890992287585488320, 6.23139342236132885671622121830, 7.00289787005198303262780691532, 7.39901460637734253545866177071, 8.292418631518924788736637322556, 9.316461492400073278794588325234

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