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Properties

Label 2-3332-476.215-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.888 + 0.459i$
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  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (0.172 + 0.349i)5-s + (−0.382 − 0.923i)8-s + (−0.130 + 0.991i)9-s + (0.349 + 0.172i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (0.130 + 0.991i)17-s + (0.499 + 0.866i)18-s + (0.382 − 0.0761i)20-s + (0.516 − 0.672i)25-s + (1.40 + 0.184i)26-s + (1.08 − 1.63i)29-s + (−0.991 + 0.130i)32-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (0.172 + 0.349i)5-s + (−0.382 − 0.923i)8-s + (−0.130 + 0.991i)9-s + (0.349 + 0.172i)10-s + (1 + i)13-s + (−0.866 − 0.499i)16-s + (0.130 + 0.991i)17-s + (0.499 + 0.866i)18-s + (0.382 − 0.0761i)20-s + (0.516 − 0.672i)25-s + (1.40 + 0.184i)26-s + (1.08 − 1.63i)29-s + (−0.991 + 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.888 + 0.459i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.888 + 0.459i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.108289484\)
\(L(\frac12)\) \(\approx\) \(2.108289484\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 \)
17 \( 1 + (-0.130 - 0.991i)T \)
good3 \( 1 + (0.130 - 0.991i)T^{2} \)
5 \( 1 + (-0.172 - 0.349i)T + (-0.608 + 0.793i)T^{2} \)
11 \( 1 + (0.793 - 0.608i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
19 \( 1 + (0.258 - 0.965i)T^{2} \)
23 \( 1 + (-0.130 - 0.991i)T^{2} \)
29 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.130 + 0.991i)T^{2} \)
37 \( 1 + (0.369 + 0.125i)T + (0.793 + 0.608i)T^{2} \)
41 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.0726 + 1.10i)T + (-0.991 + 0.130i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (1.95 + 0.128i)T + (0.991 + 0.130i)T^{2} \)
79 \( 1 + (0.130 + 0.991i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.517 + 1.93i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \)
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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

−8.745712892449942016960281711930, −8.088577712661032595631954411787, −7.04723330871851623549810227777, −6.17532371814183752241438757388, −5.88317329842079902617315750256, −4.59894307370586195686810746612, −4.25590489700342570475056028955, −3.12850615524826869405293234890, −2.29823905386918461956833042873, −1.41372263364166987020435994159, 1.14670767376512373667812128755, 2.80890978807772777775290350072, 3.38616985601162661960103332790, 4.26066085094406741014147469194, 5.31862483969887733150500408794, 5.63386625672489376936726078538, 6.65836136231685607255199045251, 7.11518356328936303258916652734, 8.093051141801003012094210377170, 8.810399242776459804913838900403

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