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Properties

Label 2-3332-476.411-c0-0-2
Degree $2$
Conductor $3332$
Sign $-0.622 + 0.783i$
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  + (0.130 − 0.991i)2-s + (−0.965 − 0.258i)4-s + (−0.0255 − 0.389i)5-s + (−0.382 + 0.923i)8-s + (0.793 − 0.608i)9-s + (−0.389 − 0.0255i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (−0.608 + 0.793i)17-s + (−0.499 − 0.866i)18-s + (−0.0761 + 0.382i)20-s + (0.840 − 0.110i)25-s + (−0.860 − 1.12i)26-s + (−0.324 + 0.216i)29-s + (0.608 − 0.793i)32-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)2-s + (−0.965 − 0.258i)4-s + (−0.0255 − 0.389i)5-s + (−0.382 + 0.923i)8-s + (0.793 − 0.608i)9-s + (−0.389 − 0.0255i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (−0.608 + 0.793i)17-s + (−0.499 − 0.866i)18-s + (−0.0761 + 0.382i)20-s + (0.840 − 0.110i)25-s + (−0.860 − 1.12i)26-s + (−0.324 + 0.216i)29-s + (0.608 − 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.622 + 0.783i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.622 + 0.783i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274313082\)
\(L(\frac12)\) \(\approx\) \(1.274313082\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 \)
17 \( 1 + (0.608 - 0.793i)T \)
good3 \( 1 + (-0.793 + 0.608i)T^{2} \)
5 \( 1 + (0.0255 + 0.389i)T + (-0.991 + 0.130i)T^{2} \)
11 \( 1 + (-0.130 + 0.991i)T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + (-0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.793 + 0.608i)T^{2} \)
29 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.793 - 0.608i)T^{2} \)
37 \( 1 + (-1.29 + 1.47i)T + (-0.130 - 0.991i)T^{2} \)
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 \( 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2} \)
59 \( 1 + (0.965 - 0.258i)T^{2} \)
61 \( 1 + (-0.491 - 0.996i)T + (-0.608 + 0.793i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (1.75 + 0.867i)T + (0.608 + 0.793i)T^{2} \)
79 \( 1 + (-0.793 - 0.608i)T^{2} \)
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

−8.778699067311722680969833270258, −8.072508069634562084428029082562, −7.07049008838614877730186898240, −6.05744567489600436833906020914, −5.43555782152308112627401244366, −4.37563164199374897496334777162, −3.85813300750595818213577474686, −2.98704459451213934769777336029, −1.78933252098970345433658068612, −0.845438977943866266081153557075, 1.40339488239027514640298440148, 2.80349223761986080950328725559, 3.88365303596113793809872778022, 4.58108621914027522718971557427, 5.23554817717457881134931217817, 6.42344234967775147378552837666, 6.69700430254564152534962938719, 7.46212816539519560917453831788, 8.246292172643835078955355875362, 8.884535617968550497803685380961

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