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Properties

Label 2-3332-476.3-c0-0-2
Degree $2$
Conductor $3332$
Sign $-0.0516 - 0.998i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0516 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0516 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.0516 - 0.998i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.0516 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.194064297\)
\(L(\frac12)\) \(\approx\) \(2.194064297\)
\(L(1)\) 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

−8.534269953946038836903667518349, −8.203613717487905045375259823897, −7.48728324717819847347101187981, −6.64891779256505004830767666400, −5.83330095445323012944140953156, −5.37205450003309324570561017798, −4.51135377191910050338141179576, −3.52798129164202713385376523399, −2.82098819551201619757154601558, −1.67117605416994708389889069628, 1.13033797369965664055026025301, 1.97529684919302614056206822078, 3.26192543483382909183639207555, 3.80214161159515946150958805452, 4.63354307393330790716066007458, 5.66555563104512051745730346163, 6.12027901003190614537550368792, 6.86918203642359167760255518047, 7.77406451797308688349958911542, 9.055666808399483117690291798219

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