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Properties

Label 2-3312-23.22-c0-0-0
Degree $2$
Conductor $3312$
Sign $1$
Analytic cond. $1.65290$
Root an. cond. $1.28565$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13-s + 23-s + 25-s + 29-s + 31-s + 41-s − 47-s + 49-s + 2·59-s − 71-s − 73-s − 2·101-s + ⋯
L(s)  = 1  − 13-s + 23-s + 25-s + 29-s + 31-s + 41-s − 47-s + 49-s + 2·59-s − 71-s − 73-s − 2·101-s + ⋯

Functional equation

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

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: yes
Arithmetic: yes
Character: $\chi_{3312} (2161, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3312,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.256416983\)
\(L(\frac12)\) \(\approx\) \(1.256416983\)
\(L(1)\) 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 \( 1 \)
23 \( 1 - T \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 - T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−8.765800349422116199227052817761, −8.142536269669701125177355953175, −7.18856710765703082079307046895, −6.75480989597837393582697331850, −5.74995401831086161401918153775, −4.91883771193027870095103284832, −4.32217118930998426829110688663, −3.09155721509330047352544334415, −2.44039303838054909941035255660, −1.02409421904395214094442928504, 1.02409421904395214094442928504, 2.44039303838054909941035255660, 3.09155721509330047352544334415, 4.32217118930998426829110688663, 4.91883771193027870095103284832, 5.74995401831086161401918153775, 6.75480989597837393582697331850, 7.18856710765703082079307046895, 8.142536269669701125177355953175, 8.765800349422116199227052817761

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