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Properties

Label 2-3332-68.67-c0-0-5
Degree $2$
Conductor $3332$
Sign $-i$
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  + 2-s + 4-s + 2i·5-s + 8-s − 9-s + 2i·10-s + 16-s + i·17-s − 18-s + 2i·20-s − 3·25-s + 32-s + i·34-s − 36-s + 2i·40-s − 2i·41-s + ⋯
L(s)  = 1  + 2-s + 4-s + 2i·5-s + 8-s − 9-s + 2i·10-s + 16-s + i·17-s − 18-s + 2i·20-s − 3·25-s + 32-s + i·34-s − 36-s + 2i·40-s − 2i·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-i$
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: \((2,\ 3332,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.214623940\)
\(L(\frac12)\) \(\approx\) \(2.214623940\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.910798471797272718739241120261, −7.932166059349246756991103152183, −7.26929135940430910527311972030, −6.63573805565276938472088701074, −5.95268570667967206761893156753, −5.47196184114791683729290094088, −4.05546430443564320995967653251, −3.50207615625483771515222682221, −2.69363529614261304227035350493, −2.06773450539787846026591481102, 0.932916404743028582333596457171, 2.10603401765731893028781930880, 3.15546472061134942580040113142, 4.19642928662406139554867024405, 4.84989405626378519022573792706, 5.42316652431438731144289533028, 5.99821582395933511915893744867, 7.04115919702666259433507616079, 8.048066470631121885324039611231, 8.430633300389028200049488011712

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