Properties

Label 2-3332-476.167-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.974 - 0.222i$
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  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.08 + 1.63i)5-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.63 + 1.08i)10-s + (−1 + i)13-s i·16-s + (−0.382 − 0.923i)17-s + 18-s + (1.92 + 0.382i)20-s + (−1.08 + 2.63i)25-s + (−0.541 + 1.30i)26-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.08 + 1.63i)5-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.63 + 1.08i)10-s + (−1 + i)13-s i·16-s + (−0.382 − 0.923i)17-s + 18-s + (1.92 + 0.382i)20-s + (−1.08 + 2.63i)25-s + (−0.541 + 1.30i)26-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.974 - 0.222i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.732340466\)
\(L(\frac12)\) \(\approx\) \(2.732340466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 \)
17 \( 1 + (0.382 + 0.923i)T \)
good3 \( 1 + (-0.923 - 0.382i)T^{2} \)
5 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (-0.923 + 0.382i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
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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

−9.436159429142385511348501309756, −7.48630868532192889407489944524, −7.23962112203092582071172051827, −6.49697655073959609450608583921, −5.88875164885539569733987236705, −4.94626846915300455026515513030, −4.19042157806563526349719489177, −3.16936810715799331158382917920, −2.25995878626728419718030403788, −1.91119600244485999334217766606, 1.36678845858280938826912420401, 2.17660335775675869784094409658, 3.43566043374277552351188392069, 4.48445383698681709180413472505, 4.91370637756199033564322352478, 5.68248807870316830422451670137, 6.23851949515265335598200283216, 7.24281537781091256121382202001, 7.936445552035025418717927714435, 8.769684432833278011057139629084

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