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Properties

Label 2-3332-476.67-c0-0-9
Degree $2$
Conductor $3332$
Sign $0.605 + 0.795i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + 0.999·8-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − 0.999·22-s + (−1 − 1.73i)23-s + (−0.499 + 0.866i)24-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)26-s − 27-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + 0.999·8-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − 0.999·22-s + (−1 − 1.73i)23-s + (−0.499 + 0.866i)24-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8026836648\)
\(L(\frac12)\) \(\approx\) \(0.8026836648\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.804391706240277141772215986494, −8.242594254273728100326010202986, −7.44967187896273881270189289222, −6.30005985605594547415842342790, −5.59359611484840553458522136793, −4.55256837653007800276850337802, −3.99925718411293761089398365115, −3.20656767476752308331533792772, −2.09350379397431204661783982583, −0.72565773226993418656413207620, 1.21504330640272431502402553812, 1.78776453573208121134216699396, 3.62515998608347669433834328421, 4.41497327763883080474903611987, 5.55630756354087906755303058675, 6.11161694034564754416256283608, 6.66089263377177453559238509445, 7.39442169777394484404913351310, 8.001163121799040453099854773457, 8.717792080139855828873757634081

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