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Properties

Label 2-3332-476.439-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.977 + 0.210i$
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 + (−1.95 + 0.128i)5-s + (0.382 − 0.923i)8-s + (−0.793 + 0.608i)9-s + (0.128 − 1.95i)10-s + (−1 + i)13-s + (0.866 + 0.5i)16-s + (−0.608 + 0.793i)17-s + (−0.499 − 0.866i)18-s + (1.92 + 0.382i)20-s + (2.82 − 0.371i)25-s + (−0.860 − 1.12i)26-s + (−1.08 − 1.63i)29-s + (−0.608 + 0.793i)32-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−0.965 − 0.258i)4-s + (−1.95 + 0.128i)5-s + (0.382 − 0.923i)8-s + (−0.793 + 0.608i)9-s + (0.128 − 1.95i)10-s + (−1 + i)13-s + (0.866 + 0.5i)16-s + (−0.608 + 0.793i)17-s + (−0.499 − 0.866i)18-s + (1.92 + 0.382i)20-s + (2.82 − 0.371i)25-s + (−0.860 − 1.12i)26-s + (−1.08 − 1.63i)29-s + (−0.608 + 0.793i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.977 + 0.210i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.977 + 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2433829916\)
\(L(\frac12)\) \(\approx\) \(0.2433829916\)
\(L(1)\) 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 \( 1 + (1.95 - 0.128i)T + (0.991 - 0.130i)T^{2} \)
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 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + (-0.793 + 0.608i)T^{2} \)
37 \( 1 + (0.293 + 0.257i)T + (0.130 + 0.991i)T^{2} \)
41 \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2} \)
59 \( 1 + (0.965 - 0.258i)T^{2} \)
61 \( 1 + (-1.49 + 0.735i)T + (0.608 - 0.793i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (-0.172 + 0.349i)T + (-0.608 - 0.793i)T^{2} \)
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 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
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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

−8.561804097382279161561649167588, −7.905310852064726178443455150062, −7.41446417555298930882076927287, −6.79992144257887720537175170017, −5.84282170873255061320412300333, −4.88039242449208098389527798715, −4.22191404972935413977719331420, −3.66228958969135815401008228742, −2.29394091253661954429630664420, −0.22147116485545379211991940325, 0.819842699255900549429978681864, 2.62453778598338894169929945115, 3.26612017678013790337823239918, 3.92360100004240393556871873787, 4.83851897787974999691799190369, 5.40338506192221491182863516561, 6.92626807436048388404239918782, 7.53503153514174717763960967800, 8.251321262513451820172794383727, 8.799582035514301676223557652427

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