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Properties

Label 2-3332-476.143-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.987 + 0.155i$
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  + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.732 + 0.835i)5-s + (0.923 − 0.382i)8-s + (0.608 − 0.793i)9-s + (0.835 + 0.732i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.608 + 0.793i)17-s + (0.499 − 0.866i)18-s + (0.923 + 0.617i)20-s + (−0.0306 + 0.232i)25-s + (−1.12 − 0.860i)26-s + (−1.63 + 0.324i)29-s + (0.793 − 0.608i)32-s + ⋯
L(s)  = 1  + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.732 + 0.835i)5-s + (0.923 − 0.382i)8-s + (0.608 − 0.793i)9-s + (0.835 + 0.732i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.608 + 0.793i)17-s + (0.499 − 0.866i)18-s + (0.923 + 0.617i)20-s + (−0.0306 + 0.232i)25-s + (−1.12 − 0.860i)26-s + (−1.63 + 0.324i)29-s + (0.793 − 0.608i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.987 + 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.796341120\)
\(L(\frac12)\) \(\approx\) \(2.796341120\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.991 + 0.130i)T \)
7 \( 1 \)
17 \( 1 + (-0.608 - 0.793i)T \)
good3 \( 1 + (-0.608 + 0.793i)T^{2} \)
5 \( 1 + (-0.732 - 0.835i)T + (-0.130 + 0.991i)T^{2} \)
11 \( 1 + (0.991 - 0.130i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + (0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.608 + 0.793i)T^{2} \)
29 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.608 - 0.793i)T^{2} \)
37 \( 1 + (1.10 + 0.0726i)T + (0.991 + 0.130i)T^{2} \)
41 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.12 - 1.46i)T + (-0.258 + 0.965i)T^{2} \)
59 \( 1 + (-0.965 - 0.258i)T^{2} \)
61 \( 1 + (1.85 - 0.630i)T + (0.793 - 0.608i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.534 - 1.57i)T + (-0.793 - 0.608i)T^{2} \)
79 \( 1 + (-0.608 - 0.793i)T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \)
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

−8.892645338485027536048518279698, −7.56041774838243003346981566123, −7.26698518013340119101130561005, −6.33452858205375881639907745451, −5.79468062844246931343882880095, −5.08044005927355055565250177523, −3.98250197887339382694531477359, −3.28775598883359752150502278112, −2.47247955608560129300163687051, −1.47356940646743944510453520689, 1.66879421853332024754131721876, 2.17319030576997066447372263369, 3.42357290203876092320910280076, 4.44728711264421563092948298680, 5.05865518881495626482606890039, 5.47754433557997530310681239218, 6.51063132445547873356034906442, 7.31365171499422328572912858374, 7.74994935162049466060882441529, 8.897657428363557508764053403321

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