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Properties

Label 2-684-171.88-c2-0-18
Degree $2$
Conductor $684$
Sign $0.985 - 0.172i$
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  + (−2.99 + 0.213i)3-s + (2.77 − 4.81i)5-s + (0.506 − 0.877i)7-s + (8.90 − 1.27i)9-s + (−3.59 + 6.21i)11-s + 21.0i·13-s + (−7.28 + 14.9i)15-s + (5.88 + 10.1i)17-s + (4.15 − 18.5i)19-s + (−1.32 + 2.73i)21-s − 38.5·23-s + (−2.93 − 5.09i)25-s + (−26.3 + 5.71i)27-s + (34.4 − 19.8i)29-s + (47.7 − 27.5i)31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0710i)3-s + (0.555 − 0.962i)5-s + (0.0723 − 0.125i)7-s + (0.989 − 0.141i)9-s + (−0.326 + 0.565i)11-s + 1.61i·13-s + (−0.485 + 0.999i)15-s + (0.346 + 0.599i)17-s + (0.218 − 0.975i)19-s + (−0.0632 + 0.130i)21-s − 1.67·23-s + (−0.117 − 0.203i)25-s + (−0.977 + 0.211i)27-s + (1.18 − 0.686i)29-s + (1.53 − 0.888i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.985 - 0.172i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.985 - 0.172i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.388177001\)
\(L(\frac12)\) \(\approx\) \(1.388177001\)
\(L(2)\) 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 \( 1 + (2.99 - 0.213i)T \)
19 \( 1 + (-4.15 + 18.5i)T \)
good5 \( 1 + (-2.77 + 4.81i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-0.506 + 0.877i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (3.59 - 6.21i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 21.0iT - 169T^{2} \)
17 \( 1 + (-5.88 - 10.1i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + 38.5T + 529T^{2} \)
29 \( 1 + (-34.4 + 19.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-47.7 + 27.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 39.2iT - 1.36e3T^{2} \)
41 \( 1 + (-1.00 - 0.580i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 - 32.6T + 1.84e3T^{2} \)
47 \( 1 + (-13.8 - 24.0i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-32.1 - 18.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-3.95 - 2.28i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.6 + 27.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 - 41.2iT - 4.48e3T^{2} \)
71 \( 1 + (-27.3 + 15.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-56.0 - 97.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + 1.05iT - 6.24e3T^{2} \)
83 \( 1 + (-76.0 + 131. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-27.7 - 16.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 136. iT - 9.40e3T^{2} \)
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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

−10.08882611190004072967091275522, −9.705798611478660419226545839350, −8.691299967323452780939034356251, −7.60913632366085480047049049409, −6.50850018580135657093769084702, −5.86705538356316118049562513144, −4.64116576375898782449169603126, −4.33467730762845243437603351752, −2.14834731932752902417908131871, −0.976047381954980540006243946507, 0.75365258000663127394507774262, 2.44348789518427692799778633289, 3.57401606586203894904248043612, 5.10335598997722537461950167379, 5.82258530513836838087046641732, 6.47295998723801684951965484845, 7.55322651630577442000261145816, 8.326833864407381043065800099341, 9.895202388343661662375885880722, 10.36350169737420970752962332771

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