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Properties

Label 2-3332-476.279-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.829 + 0.558i$
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  + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−1.08 + 0.216i)5-s + (0.923 − 0.382i)8-s + (0.382 + 0.923i)9-s + (0.216 − 1.08i)10-s + (−1 − i)13-s + i·16-s + (0.382 − 0.923i)17-s − 18-s + (0.923 + 0.617i)20-s + (0.216 − 0.0897i)25-s + (1.30 − 0.541i)26-s + (−1.63 + 0.324i)29-s + (−0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.707 − 0.707i)4-s + (−1.08 + 0.216i)5-s + (0.923 − 0.382i)8-s + (0.382 + 0.923i)9-s + (0.216 − 1.08i)10-s + (−1 − i)13-s + i·16-s + (0.382 − 0.923i)17-s − 18-s + (0.923 + 0.617i)20-s + (0.216 − 0.0897i)25-s + (1.30 − 0.541i)26-s + (−1.63 + 0.324i)29-s + (−0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.829 + 0.558i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.829 + 0.558i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4658608029\)
\(L(\frac12)\) \(\approx\) \(0.4658608029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 \)
17 \( 1 + (-0.382 + 0.923i)T \)
good3 \( 1 + (-0.382 - 0.923i)T^{2} \)
5 \( 1 + (1.08 - 0.216i)T + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.382 - 0.923i)T^{2} \)
29 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (-1.63 + 0.324i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
97 \( 1 + (0.324 + 1.63i)T + (-0.923 + 0.382i)T^{2} \)
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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

−8.358411419394985777200342971801, −7.88219996787708898077169743107, −7.34451318918079184160997367945, −6.89907983964762003195097550129, −5.53069892173393912622571144821, −5.16234477440234487098066807691, −4.25773054531528509053914541462, −3.35843401669977870108536445147, −2.05701410331284451040445214289, −0.36059908334400582422416166607, 1.17028550969851804372648981741, 2.30545358562821348702677656642, 3.47335404201374123687714644612, 4.06550273769375665735212058696, 4.59780246802268836612957405555, 5.80104319044284551948325549871, 6.96261564449593814341954809725, 7.53432950095458871578746403638, 8.235067267082407739747461925078, 9.013681230999474553388417426687

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