Properties

Label 2-3332-68.67-c0-0-15
Degree $2$
Conductor $3332$
Sign $1$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + 2-s + 3-s + 4-s + 6-s + 8-s − 11-s + 12-s + 13-s + 16-s − 17-s − 22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·31-s + 32-s − 33-s − 34-s + 39-s − 44-s + 2·46-s + 48-s + 50-s − 51-s + 52-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s − 11-s + 12-s + 13-s + 16-s − 17-s − 22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s − 2·31-s + 32-s − 33-s − 34-s + 39-s − 44-s + 2·46-s + 48-s + 50-s − 51-s + 52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.238767036\)
\(L(\frac12)\) \(\approx\) \(3.238767036\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
17 \( 1 + T \)
good3 \( 1 - T + T^{2} \)
5 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T + T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 + T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 - T + T^{2} \)
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

−8.802458348725255232615498550120, −8.000561748114039096900652839120, −7.26181370817594147845769564814, −6.59949655405820058488780064255, −5.60066543217778880257595535868, −4.98885668953084857028779377323, −4.01888448090175434953144994679, −3.15952568315949717571176898170, −2.67549620888112336507036147811, −1.60719134774815446283767953176, 1.60719134774815446283767953176, 2.67549620888112336507036147811, 3.15952568315949717571176898170, 4.01888448090175434953144994679, 4.98885668953084857028779377323, 5.60066543217778880257595535868, 6.59949655405820058488780064255, 7.26181370817594147845769564814, 8.000561748114039096900652839120, 8.802458348725255232615498550120

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