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Properties

Label 2-3312-3312.229-c0-0-2
Degree $2$
Conductor $3312$
Sign $0.976 + 0.216i$
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 + (−0.939 − 0.342i)3-s + (0.939 − 0.342i)4-s + (−0.984 − 0.173i)6-s + (0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s − 0.999·12-s + (0.469 + 1.75i)13-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)18-s + (0.866 + 0.5i)23-s + (−0.984 + 0.173i)24-s + (0.866 − 0.5i)25-s + (0.766 + 1.64i)26-s + (−0.500 − 0.866i)27-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (−0.939 − 0.342i)3-s + (0.939 − 0.342i)4-s + (−0.984 − 0.173i)6-s + (0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s − 0.999·12-s + (0.469 + 1.75i)13-s + (0.766 − 0.642i)16-s + (0.866 + 0.5i)18-s + (0.866 + 0.5i)23-s + (−0.984 + 0.173i)24-s + (0.866 − 0.5i)25-s + (0.766 + 1.64i)26-s + (−0.500 − 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $0.976 + 0.216i$
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: \((2,\ 3312,\ (\ :0),\ 0.976 + 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896590788\)
\(L(\frac12)\) \(\approx\) \(1.896590788\)
\(L(1)\) 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 \( 1 + (0.939 + 0.342i)T \)
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 \( 1 + (-0.469 - 1.75i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (0.984 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
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

−8.862074471764356597228816452977, −7.68224453683818519871647600048, −6.80906537415791487485426499387, −6.67112989524340421976209957520, −5.64432831804098891918820089147, −5.01136408064950526960681909915, −4.27542362660882072484888077361, −3.44373007406499077822169576533, −2.12287950163953402234142490362, −1.34435125944067868203580256593, 1.09723254902610118301946448378, 2.62042068429151290142417782339, 3.53504809596254844015910258101, 4.26823217502282054340518328052, 5.22676534734122191703039167939, 5.65640651140246847742693261968, 6.27891090757830337672384870875, 7.27498700671766166006633908403, 7.71092656255361950546509248379, 8.827924095001073421434939706586

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