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Properties

Label 2-3332-476.219-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.782 + 0.622i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.0999 + 0.758i)5-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.0999 + 0.758i)10-s + (0.500 − 0.866i)16-s + (0.258 − 0.965i)17-s + (−0.499 − 0.866i)18-s + (0.292 + 0.707i)20-s + (0.400 + 0.107i)25-s + (0.707 − 0.292i)29-s + (0.258 − 0.965i)32-s − i·34-s + (−0.707 − 0.707i)36-s + (−0.758 − 0.0999i)37-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.0999 + 0.758i)5-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (0.0999 + 0.758i)10-s + (0.500 − 0.866i)16-s + (0.258 − 0.965i)17-s + (−0.499 − 0.866i)18-s + (0.292 + 0.707i)20-s + (0.400 + 0.107i)25-s + (0.707 − 0.292i)29-s + (0.258 − 0.965i)32-s − i·34-s + (−0.707 − 0.707i)36-s + (−0.758 − 0.0999i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.782 + 0.622i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.782 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.337929726\)
\(L(\frac12)\) \(\approx\) \(2.337929726\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 \)
17 \( 1 + (-0.258 + 0.965i)T \)
good3 \( 1 + (0.258 + 0.965i)T^{2} \)
5 \( 1 + (0.0999 - 0.758i)T + (-0.965 - 0.258i)T^{2} \)
11 \( 1 + (-0.965 + 0.258i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.258 - 0.965i)T^{2} \)
29 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.258 + 0.965i)T^{2} \)
37 \( 1 + (0.758 + 0.0999i)T + (0.965 + 0.258i)T^{2} \)
41 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.46 - 1.12i)T + (0.258 - 0.965i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2} \)
79 \( 1 + (0.258 - 0.965i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
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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

−8.846992082107129294991991572468, −7.64462459593075014932910858921, −7.08963210481104696526145625933, −6.34833699634909290996655218652, −5.78574042535382540488472784667, −4.79784127246536892237645421939, −4.00611243688046569265829950770, −3.05881403288902208041231194044, −2.65978273427005856408934760394, −1.14664259525085344940689738437, 1.52970875803273112050047386594, 2.51625649900098086544564438285, 3.52904739595239383353224089799, 4.42912036652494299158451558253, 5.03929175683825649920556231585, 5.69521207892212061587891396949, 6.50268984302776428760915017593, 7.36841971754859884837870252907, 8.168844529746529391380417264197, 8.508092881689888240472339732651

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