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Properties

Label 2-3312-3312.229-c0-0-5
Degree $2$
Conductor $3312$
Sign $-0.887 + 0.461i$
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  + (−0.984 − 0.173i)2-s + (−0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (0.5 + 0.866i)6-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.342 − 0.939i)12-s + (−0.515 − 1.92i)13-s + (0.766 + 0.642i)16-s + (0.342 − 0.939i)18-s + (0.866 + 0.5i)23-s + (0.173 + 0.984i)24-s + (0.866 − 0.5i)25-s + (0.173 + 1.98i)26-s + (0.866 − 0.500i)27-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (0.5 + 0.866i)6-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.342 − 0.939i)12-s + (−0.515 − 1.92i)13-s + (0.766 + 0.642i)16-s + (0.342 − 0.939i)18-s + (0.866 + 0.5i)23-s + (0.173 + 0.984i)24-s + (0.866 − 0.5i)25-s + (0.173 + 1.98i)26-s + (0.866 − 0.500i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.887 + 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4167082226\)
\(L(\frac12)\) \(\approx\) \(0.4167082226\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (0.642 + 0.766i)T \)
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 \( 1 + (0.515 + 1.92i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (1.75 + 0.469i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.342 - 0.592i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 0.684iT - T^{2} \)
73 \( 1 + 1.87iT - T^{2} \)
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

−8.411285123028505988126225880520, −7.52981241654370177130548405460, −7.42079887122170255337092358563, −6.36200978314868696729823266960, −5.66131382341309420547053539134, −4.95670071345771556008734598675, −3.38770742554482498857053725630, −2.62163458090357182896465936179, −1.53596977511812618683852901094, −0.39261782088412013666598002867, 1.36701744377852496819475192234, 2.53233405679951764178057777189, 3.70095501583847605992055127639, 4.63728693417329250687141168032, 5.41854384180849866958143587265, 6.29507835721447010102461034178, 6.93666703013705250240134945769, 7.51710366674269232053820849342, 8.740242123131511809479603640483, 9.289839632530095976560867218260

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