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Properties

Label 2-3312-207.22-c0-0-1
Degree $2$
Conductor $3312$
Sign $1$
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  + (0.939 + 0.342i)3-s + (0.766 + 0.642i)9-s + (0.939 − 1.62i)13-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.500 + 0.866i)27-s + (−0.173 − 0.300i)29-s + (0.173 − 0.300i)31-s + (1.43 − 1.20i)39-s + (−0.766 + 1.32i)41-s + (0.766 + 1.32i)47-s + (−0.5 + 0.866i)49-s + (−0.5 + 0.866i)59-s + (0.766 − 0.642i)69-s − 0.347·71-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)3-s + (0.766 + 0.642i)9-s + (0.939 − 1.62i)13-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.500 + 0.866i)27-s + (−0.173 − 0.300i)29-s + (0.173 − 0.300i)31-s + (1.43 − 1.20i)39-s + (−0.766 + 1.32i)41-s + (0.766 + 1.32i)47-s + (−0.5 + 0.866i)49-s + (−0.5 + 0.866i)59-s + (0.766 − 0.642i)69-s − 0.347·71-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3312\)    =    \(2^{4} \cdot 3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1.65290\)
Root analytic conductor: \(1.28565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3312} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.869791826\)
\(L(\frac12)\) \(\approx\) \(1.869791826\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + 0.347T + T^{2} \)
73 \( 1 - 0.347T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−8.747628589069992566145807940433, −8.025162363964493541511551352562, −7.72142841579130517581123156505, −6.54118169006675636896898429275, −5.82793212095022638470309336204, −4.82913555189173550806650143482, −4.07840462782250195333438567221, −3.15132058807160964189710449331, −2.56058687177002190591109350044, −1.19595653589390158223348912010, 1.45426631760614867826947508116, 2.11564951866663963784259987469, 3.44376244381988360942480114083, 3.82967319322874241211766562271, 4.90284248535422175694123211522, 5.91784289283312042937136978267, 6.88243393139957331942768361825, 7.18322171449098925656216085381, 8.195318412665393463577684439726, 8.862011166630680930342202965799

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