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Properties

Label 2-684-171.7-c1-0-14
Degree $2$
Conductor $684$
Sign $0.833 + 0.552i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71 − 0.221i)3-s + (1.53 − 2.65i)5-s + (−2.37 + 4.11i)7-s + (2.90 − 0.761i)9-s + (2.85 − 4.94i)11-s − 1.84·13-s + (2.04 − 4.89i)15-s + (0.661 + 1.14i)17-s + (4.34 − 0.301i)19-s + (−3.16 + 7.58i)21-s + 4.21·23-s + (−2.18 − 3.78i)25-s + (4.81 − 1.95i)27-s + (−3.26 − 5.64i)29-s + (1.80 + 3.11i)31-s + ⋯
L(s)  = 1  + (0.991 − 0.127i)3-s + (0.684 − 1.18i)5-s + (−0.897 + 1.55i)7-s + (0.967 − 0.253i)9-s + (0.860 − 1.49i)11-s − 0.512·13-s + (0.527 − 1.26i)15-s + (0.160 + 0.277i)17-s + (0.997 − 0.0692i)19-s + (−0.690 + 1.65i)21-s + 0.879·23-s + (−0.437 − 0.757i)25-s + (0.926 − 0.375i)27-s + (−0.605 − 1.04i)29-s + (0.323 + 0.560i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.833 + 0.552i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.833 + 0.552i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18251 - 0.657241i\)
\(L(\frac12)\) \(\approx\) \(2.18251 - 0.657241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.71 + 0.221i)T \)
19 \( 1 + (-4.34 + 0.301i)T \)
good5 \( 1 + (-1.53 + 2.65i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.37 - 4.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.85 + 4.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.84T + 13T^{2} \)
17 \( 1 + (-0.661 - 1.14i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 - 4.21T + 23T^{2} \)
29 \( 1 + (3.26 + 5.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.80 - 3.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.97T + 37T^{2} \)
41 \( 1 + (4.64 - 8.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.541T + 43T^{2} \)
47 \( 1 + (4.65 + 8.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.93 - 8.54i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.92 - 3.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.777 + 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 + (-0.238 - 0.412i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.83 - 13.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + (5.22 - 9.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.49 + 2.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.7T + 97T^{2} \)
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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

−9.809557482576113591368996739171, −9.425173720012723860486994094141, −8.729713414070492460234571892714, −8.277700351331009449178480797198, −6.75985328546493818408823299665, −5.85344760198237686660987990014, −5.08242925847696130715774637636, −3.53436574689976992113812886504, −2.66883751578453844860523394128, −1.30336734306948371335698879193, 1.68740921374336496543374140312, 3.02144804605646199311944540786, 3.72337310506451730095685557714, 4.86364785748866341628176849281, 6.64994017981371485064946323631, 7.05688903613262483489168448369, 7.56815163608927462765263288905, 9.228672718181957540693037481522, 9.830558577584006267405142536075, 10.16403901305381671224494588059

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