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Properties

Label 2-3344-209.113-c0-0-2
Degree $2$
Conductor $3344$
Sign $-0.520 + 0.853i$
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  + (−1.08 − 0.786i)5-s + (0.604 − 1.86i)7-s + (−0.809 + 0.587i)9-s + (0.978 + 0.207i)11-s + (0.169 + 0.122i)17-s + (−0.309 − 0.951i)19-s + 1.61·23-s + (0.244 + 0.752i)25-s + (−2.11 + 1.53i)35-s − 1.82·43-s + 1.33·45-s + (−0.564 − 1.73i)47-s + (−2.28 − 1.66i)49-s + (−0.895 − 0.994i)55-s + (−1.47 − 1.07i)61-s + ⋯
L(s)  = 1  + (−1.08 − 0.786i)5-s + (0.604 − 1.86i)7-s + (−0.809 + 0.587i)9-s + (0.978 + 0.207i)11-s + (0.169 + 0.122i)17-s + (−0.309 − 0.951i)19-s + 1.61·23-s + (0.244 + 0.752i)25-s + (−2.11 + 1.53i)35-s − 1.82·43-s + 1.33·45-s + (−0.564 − 1.73i)47-s + (−2.28 − 1.66i)49-s + (−0.895 − 0.994i)55-s + (−1.47 − 1.07i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.520 + 0.853i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ -0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9206469774\)
\(L(\frac12)\) \(\approx\) \(0.9206469774\)
\(L(1)\) 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 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.604 + 1.86i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 1.82T + T^{2} \)
47 \( 1 + (0.564 + 1.73i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−8.432609291123674831011633966613, −7.88483645804203532468857882450, −7.11273981434505145063373661231, −6.63393810289399928232993287129, −5.08428192319744946142618074515, −4.73546346409650464176209522892, −3.95444105883581632061226860230, −3.23686678247464604980806526753, −1.60700824577877602044701426920, −0.58186343141639475249087178354, 1.60492483347314510242363251161, 2.95985468004343753569168368029, 3.24378231808537634540086460343, 4.41243839913634709264469338502, 5.35863049015586802979633022161, 6.12137016223245840258566696333, 6.68905744821266697586951644499, 7.73088521226123193946071886952, 8.359351660739090040601690298782, 8.960980374399945861775385655996

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