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Properties

Label 2-3312-3312.1885-c0-0-4
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  + (0.342 − 0.939i)2-s + (0.766 − 0.642i)3-s + (−0.766 − 0.642i)4-s + (−0.342 − 0.939i)6-s + (−0.866 + 0.500i)8-s + (0.173 − 0.984i)9-s − 12-s + (1.92 − 0.515i)13-s + (0.173 + 0.984i)16-s + (−0.866 − 0.499i)18-s + (−0.866 − 0.5i)23-s + (−0.342 + 0.939i)24-s + (−0.866 + 0.5i)25-s + (0.173 − 1.98i)26-s + (−0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (0.766 − 0.642i)3-s + (−0.766 − 0.642i)4-s + (−0.342 − 0.939i)6-s + (−0.866 + 0.500i)8-s + (0.173 − 0.984i)9-s − 12-s + (1.92 − 0.515i)13-s + (0.173 + 0.984i)16-s + (−0.866 − 0.499i)18-s + (−0.866 − 0.5i)23-s + (−0.342 + 0.939i)24-s + (−0.866 + 0.5i)25-s + (0.173 − 1.98i)26-s + (−0.500 − 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: $\chi_{3312} (1885, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ -0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.821373214\)
\(L(\frac12)\) \(\approx\) \(1.821373214\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
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 \( 1 + (-1.92 + 0.515i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
29 \( 1 + (-0.469 + 1.75i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.866 + 0.5i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - 0.684iT - T^{2} \)
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

−8.322372370796855374657597346709, −8.276582304833239671695287388464, −6.99093457977111578244051613997, −6.06125406338824025186410119012, −5.62143198268188153495937441902, −4.13636820718198146949061644606, −3.76334066813434471756356124782, −2.81664058470286099053407428894, −1.92692716769909932201509532112, −0.962974797221329024648687633866, 1.73432465665304262546857169092, 3.16323724246079378497562993558, 3.77703197822729686611435607548, 4.40266400060797784307658354651, 5.35818183146252316554547595208, 6.11192030217472055074048589682, 6.82551538962771602017880825750, 7.87668094284035909926629422691, 8.227032280987991065995174448110, 9.039487680986798238585460868853

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