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Properties

Label 2-3312-3312.1931-c0-0-0
Degree $2$
Conductor $3312$
Sign $-0.843 - 0.537i$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
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  + (−0.642 + 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 − 0.984i)4-s + (−0.5 − 0.866i)6-s + (0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (0.984 + 0.173i)12-s + (−0.218 − 0.816i)13-s + (−0.939 + 0.342i)16-s + (0.984 − 0.173i)18-s + (0.866 + 0.5i)23-s + (−0.766 + 0.642i)24-s + (−0.866 + 0.5i)25-s + (0.766 + 0.357i)26-s + (0.866 − 0.500i)27-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.342 + 0.939i)3-s + (−0.173 − 0.984i)4-s + (−0.5 − 0.866i)6-s + (0.866 + 0.500i)8-s + (−0.766 − 0.642i)9-s + (0.984 + 0.173i)12-s + (−0.218 − 0.816i)13-s + (−0.939 + 0.342i)16-s + (0.984 − 0.173i)18-s + (0.866 + 0.5i)23-s + (−0.766 + 0.642i)24-s + (−0.866 + 0.5i)25-s + (0.766 + 0.357i)26-s + (0.866 − 0.500i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $-0.843 - 0.537i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (1931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.843 - 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6145362839\)
\(L(\frac12)\) \(\approx\) \(0.6145362839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (0.218 + 0.816i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (0.296 - 1.10i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (0.642 - 1.11i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.96iT - T^{2} \)
73 \( 1 - 0.347iT - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.120164454362872640644013748392, −8.461646020987468558422312566364, −7.68347091464873621508536417872, −6.91090086308907449160129754377, −6.04319545906061158512612775509, −5.36900882382608198160653056039, −4.83725303289387563419040679784, −3.79996058667943677746346068228, −2.77732117185463001323812147820, −1.18328137159075420159996799257, 0.53665236350850156341384350860, 1.88158949536688140615098768839, 2.42433281159642997527041669348, 3.60859214128022210813365854996, 4.53839698936558276829755782350, 5.51501585504477563643003237917, 6.56544622629522023494284993594, 7.09160763754875871218739752624, 7.85790345559402042397587096104, 8.526356929619930760333609225079

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