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Properties

Label 2-3312-3312.3035-c0-0-2
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 + (−0.173 + 0.984i)3-s + (−0.173 − 0.984i)4-s + (0.642 + 0.766i)6-s + (−0.866 − 0.500i)8-s + (−0.939 − 0.342i)9-s + 12-s + (1.10 + 0.296i)13-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.499i)18-s + (0.866 − 0.5i)23-s + (0.642 − 0.766i)24-s + (0.866 + 0.5i)25-s + (0.939 − 0.657i)26-s + (0.5 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.173 + 0.984i)3-s + (−0.173 − 0.984i)4-s + (0.642 + 0.766i)6-s + (−0.866 − 0.500i)8-s + (−0.939 − 0.342i)9-s + 12-s + (1.10 + 0.296i)13-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.499i)18-s + (0.866 − 0.5i)23-s + (0.642 − 0.766i)24-s + (0.866 + 0.5i)25-s + (0.939 − 0.657i)26-s + (0.5 − 0.866i)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: $\chi_{3312} (3035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.649966412\)
\(L(\frac12)\) \(\approx\) \(1.649966412\)
\(L(1)\) 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 \( 1 + (0.173 - 0.984i)T \)
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 \( 1 + (-1.10 - 0.296i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (-1.92 + 0.515i)T + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.342 - 0.592i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.28iT - T^{2} \)
73 \( 1 - 1.53iT - T^{2} \)
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

−8.734460167906009077212541358247, −8.561456703750401046972681203224, −6.89814231775686922666542764509, −6.33161507086063701725152319208, −5.36292812622243039693864045198, −4.88463856693998103027132236823, −3.96743317855288892554877128291, −3.37210176869718678526041075869, −2.47777474382196567813377801694, −1.06024097941657723006823097044, 1.14768912412170124091201062165, 2.60162569201937901423397161921, 3.30991501860175034690295972423, 4.41520553913338726126143245474, 5.27779884313398934544837271974, 6.04430980264082726110928292432, 6.50883273878321528531298535633, 7.37273136244538879184286243980, 7.83939469496775769556172285713, 8.785395514039125356905528534695

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