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Properties

Label 2-684-171.103-c2-0-17
Degree $2$
Conductor $684$
Sign $0.920 - 0.391i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.94 − 0.573i)3-s + (1.51 + 2.63i)5-s + (−4.58 − 7.94i)7-s + (8.34 − 3.37i)9-s + (8.59 + 14.8i)11-s + 16.7i·13-s + (5.97 + 6.87i)15-s + (−2.67 + 4.62i)17-s + (5.04 + 18.3i)19-s + (−18.0 − 20.7i)21-s + 3.30·23-s + (7.88 − 13.6i)25-s + (22.6 − 14.7i)27-s + (43.0 + 24.8i)29-s + (−22.3 − 12.8i)31-s + ⋯
L(s)  = 1  + (0.981 − 0.191i)3-s + (0.303 + 0.526i)5-s + (−0.655 − 1.13i)7-s + (0.927 − 0.374i)9-s + (0.781 + 1.35i)11-s + 1.28i·13-s + (0.398 + 0.458i)15-s + (−0.157 + 0.272i)17-s + (0.265 + 0.964i)19-s + (−0.860 − 0.989i)21-s + 0.143·23-s + (0.315 − 0.546i)25-s + (0.838 − 0.545i)27-s + (1.48 + 0.857i)29-s + (−0.720 − 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.758917766\)
\(L(\frac12)\) \(\approx\) \(2.758917766\)
\(L(2)\) 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 \( 1 + (-2.94 + 0.573i)T \)
19 \( 1 + (-5.04 - 18.3i)T \)
good5 \( 1 + (-1.51 - 2.63i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.58 + 7.94i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.59 - 14.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 16.7iT - 169T^{2} \)
17 \( 1 + (2.67 - 4.62i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 3.30T + 529T^{2} \)
29 \( 1 + (-43.0 - 24.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (22.3 + 12.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 66.3iT - 1.36e3T^{2} \)
41 \( 1 + (-62.8 + 36.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 31.5T + 1.84e3T^{2} \)
47 \( 1 + (17.1 - 29.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (9.05 - 5.22i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (6.59 - 3.80i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-3.93 + 6.81i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 106. iT - 4.48e3T^{2} \)
71 \( 1 + (63.9 + 36.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (4.59 - 7.95i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 129. iT - 6.24e3T^{2} \)
83 \( 1 + (62.6 + 108. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (98.3 - 56.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 27.3iT - 9.40e3T^{2} \)
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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

−10.10044206039825026911240223365, −9.541086680613396235818940216898, −8.773259393638818221843565752140, −7.35757375972858937543947445255, −7.08608335206828038462957826850, −6.23858248195363745514039070110, −4.30044590057738583519064303516, −3.89788023737194828740545039178, −2.50744434876830024351704753906, −1.40131917707318348429200762158, 1.01265275567153982304150274890, 2.75004530512010217995079014754, 3.22451198515677966558246918011, 4.71697925838955123770937844808, 5.71565941235535837238579193157, 6.59311169324889055108560332143, 7.924591410016689883917068285437, 8.722780361381548786548911633934, 9.148984419909583972944927106692, 9.928204487720827356369567242508

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