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Properties

Label 2-3332-476.159-c0-0-3
Degree $2$
Conductor $3332$
Sign $-0.864 + 0.502i$
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  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.349 + 0.172i)5-s + (−0.382 − 0.923i)8-s + (0.130 − 0.991i)9-s + (−0.172 + 0.349i)10-s + (−1 − i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (−0.499 − 0.866i)18-s + (0.0761 + 0.382i)20-s + (−0.516 + 0.672i)25-s + (−1.40 − 0.184i)26-s + (−0.324 − 0.216i)29-s + (−0.991 + 0.130i)32-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.349 + 0.172i)5-s + (−0.382 − 0.923i)8-s + (0.130 − 0.991i)9-s + (−0.172 + 0.349i)10-s + (−1 − i)13-s + (−0.866 − 0.499i)16-s + (−0.991 + 0.130i)17-s + (−0.499 − 0.866i)18-s + (0.0761 + 0.382i)20-s + (−0.516 + 0.672i)25-s + (−1.40 − 0.184i)26-s + (−0.324 − 0.216i)29-s + (−0.991 + 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.864 + 0.502i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.864 + 0.502i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.403391968\)
\(L(\frac12)\) \(\approx\) \(1.403391968\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 \)
17 \( 1 + (0.991 - 0.130i)T \)
good3 \( 1 + (-0.130 + 0.991i)T^{2} \)
5 \( 1 + (0.349 - 0.172i)T + (0.608 - 0.793i)T^{2} \)
11 \( 1 + (-0.793 + 0.608i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (0.258 - 0.965i)T^{2} \)
23 \( 1 + (0.130 + 0.991i)T^{2} \)
29 \( 1 + (0.324 + 0.216i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.130 - 0.991i)T^{2} \)
37 \( 1 + (-0.630 + 1.85i)T + (-0.793 - 0.608i)T^{2} \)
41 \( 1 + (-1.38 + 0.923i)T + (0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (-1.10 + 0.0726i)T + (0.991 - 0.130i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T^{2} \)
73 \( 1 + (0.128 - 1.95i)T + (-0.991 - 0.130i)T^{2} \)
79 \( 1 + (-0.130 - 0.991i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
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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

−8.662704941939732012173802744626, −7.43713737851935356686464104085, −7.05957275007627465701454600804, −5.95718901891157902315406312740, −5.51850515245026087390320632188, −4.37892590356150507982910794288, −3.86056835491549744986174093378, −2.92822487682543696017621072013, −2.10837191417237831268563741099, −0.59823070558453120155209330107, 2.01713412011958660145452404471, 2.72461772779227936799781960889, 3.98570685806704546675783918055, 4.64317282190535450903510474883, 5.06438398969168336197027458874, 6.22105422290068290969695633299, 6.79415029912883587373724708563, 7.63271119340785460147728478656, 8.069519295141987841187839695586, 8.932512137357172926157696315218

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