Properties

Label 2-3332-476.199-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.0799 + 0.996i$
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  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (1.57 − 0.534i)5-s + (−0.923 − 0.382i)8-s + (0.991 − 0.130i)9-s + (0.534 − 1.57i)10-s + (−1 + i)13-s + (−0.866 + 0.499i)16-s + (0.991 + 0.130i)17-s + (0.499 − 0.866i)18-s + (−0.923 − 1.38i)20-s + (1.40 − 1.07i)25-s + (0.184 + 1.40i)26-s + (0.216 − 1.08i)29-s + (−0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (1.57 − 0.534i)5-s + (−0.923 − 0.382i)8-s + (0.991 − 0.130i)9-s + (0.534 − 1.57i)10-s + (−1 + i)13-s + (−0.866 + 0.499i)16-s + (0.991 + 0.130i)17-s + (0.499 − 0.866i)18-s + (−0.923 − 1.38i)20-s + (1.40 − 1.07i)25-s + (0.184 + 1.40i)26-s + (0.216 − 1.08i)29-s + (−0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0799 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0799 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.0799 + 0.996i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.0799 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.285476351\)
\(L(\frac12)\) \(\approx\) \(2.285476351\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 \)
17 \( 1 + (-0.991 - 0.130i)T \)
good3 \( 1 + (-0.991 + 0.130i)T^{2} \)
5 \( 1 + (-1.57 + 0.534i)T + (0.793 - 0.608i)T^{2} \)
11 \( 1 + (0.608 - 0.793i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (-0.258 - 0.965i)T^{2} \)
23 \( 1 + (0.991 + 0.130i)T^{2} \)
29 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.991 - 0.130i)T^{2} \)
37 \( 1 + (1.49 + 0.735i)T + (0.608 + 0.793i)T^{2} \)
41 \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T^{2} \)
61 \( 1 + (-0.257 - 0.293i)T + (-0.130 + 0.991i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.835 + 0.732i)T + (0.130 + 0.991i)T^{2} \)
79 \( 1 + (-0.991 - 0.130i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
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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

−9.108839019904040536728308506684, −7.897611012187262305253345812752, −6.78685895955099553901549777303, −6.23403167488140592312474441663, −5.37019024730938445621440473390, −4.78746560354967118010038731683, −4.04816997770593960206147765767, −2.82798041888949857726112042234, −1.91841852709683010500346507431, −1.31989072699075940030469303628, 1.67096847396154277911218242612, 2.74084307522887046437466154344, 3.44527533878281614325521235279, 4.72514025068184660637238381208, 5.35870789255967555539216356582, 5.85177288685282521084676760122, 6.88204261696370832228427210807, 7.17455044995273549527328838393, 8.042000406890544837980758582651, 9.005031086838998283917851650280

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