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Properties

Label 2-3332-68.67-c0-0-14
Degree $2$
Conductor $3332$
Sign $0.951 + 0.309i$
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  + (−0.809 + 0.587i)2-s + 1.17·3-s + (0.309 − 0.951i)4-s − 0.618i·5-s + (−0.951 + 0.690i)6-s + (0.309 + 0.951i)8-s + 0.381·9-s + (0.363 + 0.500i)10-s + (0.363 − 1.11i)12-s − 0.726i·15-s + (−0.809 − 0.587i)16-s − i·17-s + (−0.309 + 0.224i)18-s + (−0.587 − 0.190i)20-s + (0.363 + 1.11i)24-s + 0.618·25-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + 1.17·3-s + (0.309 − 0.951i)4-s − 0.618i·5-s + (−0.951 + 0.690i)6-s + (0.309 + 0.951i)8-s + 0.381·9-s + (0.363 + 0.500i)10-s + (0.363 − 1.11i)12-s − 0.726i·15-s + (−0.809 − 0.587i)16-s − i·17-s + (−0.309 + 0.224i)18-s + (−0.587 − 0.190i)20-s + (0.363 + 1.11i)24-s + 0.618·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.951 + 0.309i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.951 + 0.309i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.257291149\)
\(L(\frac12)\) \(\approx\) \(1.257291149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 \)
17 \( 1 + iT \)
good3 \( 1 - 1.17T + T^{2} \)
5 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.90T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.61iT - T^{2} \)
43 \( 1 + 1.17iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.61iT - T^{2} \)
67 \( 1 - 1.90iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.61iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 0.618iT - T^{2} \)
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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

−8.857097178353381283242253926760, −8.205877608702363519765041629417, −7.44132914908622607486502825226, −6.87365651145692853744920416576, −5.80982575855188086152550567744, −5.06568078194350007527997971312, −4.16917354785186852466251742611, −2.92359918632938164209394546321, −2.19560185736846533075566238329, −0.925432228531517379927853493944, 1.39310252268959699561528607274, 2.47523039157240254767747154150, 3.01440043748345871667891083389, 3.75827565334714500279100061670, 4.69323461868182456792323501793, 6.22689235162988796122546001786, 6.76192559426606521218222620902, 7.84963236262360247971500578997, 8.116658035477302405843916813163, 8.793044901103996014908936451979

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