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Properties

Label 2-3332-476.311-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.895 - 0.445i$
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 + (−0.534 − 1.57i)5-s + (−0.923 − 0.382i)8-s + (−0.991 + 0.130i)9-s + (−1.57 − 0.534i)10-s + (1 − i)13-s + (−0.866 + 0.499i)16-s + (0.130 − 0.991i)17-s + (−0.499 + 0.866i)18-s + (−1.38 + 0.923i)20-s + (−1.40 + 1.07i)25-s + (−0.184 − 1.40i)26-s + (1.63 + 0.324i)29-s + (−0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (−0.534 − 1.57i)5-s + (−0.923 − 0.382i)8-s + (−0.991 + 0.130i)9-s + (−1.57 − 0.534i)10-s + (1 − i)13-s + (−0.866 + 0.499i)16-s + (0.130 − 0.991i)17-s + (−0.499 + 0.866i)18-s + (−1.38 + 0.923i)20-s + (−1.40 + 1.07i)25-s + (−0.184 − 1.40i)26-s + (1.63 + 0.324i)29-s + (−0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.895 - 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141354114\)
\(L(\frac12)\) \(\approx\) \(1.141354114\)
\(L(1)\) 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 \( 1 + (-0.130 + 0.991i)T \)
good3 \( 1 + (0.991 - 0.130i)T^{2} \)
5 \( 1 + (0.534 + 1.57i)T + (-0.793 + 0.608i)T^{2} \)
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 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.991 + 0.130i)T^{2} \)
37 \( 1 + (0.491 - 0.996i)T + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.293 + 0.257i)T + (0.130 - 0.991i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.732 + 0.835i)T + (-0.130 - 0.991i)T^{2} \)
79 \( 1 + (0.991 + 0.130i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
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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

−8.424952072289677369726124731143, −8.073454637115453587013015163165, −6.64164130737538146996194033640, −5.79976611242760348588671860021, −4.97663564834756204359465793317, −4.77239086414571707326748647160, −3.50088757675963153961695633495, −2.98079455334681680331335778326, −1.53242877488487546743621774221, −0.56080221372735873235642345457, 2.20955539447899786613584365787, 3.27250105036722806385893756536, 3.62890746267585568615261927355, 4.58265264158159692164843806250, 5.73459029149542614056612855745, 6.42652974503423875727153413366, 6.70824301231600684292269692448, 7.62444799523350955540845546439, 8.390334337997113082254271267978, 8.808052453010583492879559592471

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