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Properties

Label 2-684-171.49-c1-0-6
Degree $2$
Conductor $684$
Sign $0.988 + 0.148i$
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  + (0.143 − 1.72i)3-s + (−0.479 − 0.830i)5-s + (2.11 + 3.66i)7-s + (−2.95 − 0.494i)9-s + (2.66 + 4.61i)11-s + 6.85·13-s + (−1.50 + 0.708i)15-s + (−2.94 + 5.10i)17-s + (−2.72 − 3.39i)19-s + (6.63 − 3.12i)21-s + 3.55·23-s + (2.04 − 3.53i)25-s + (−1.27 + 5.03i)27-s + (−2.13 + 3.69i)29-s + (2.25 − 3.90i)31-s + ⋯
L(s)  = 1  + (0.0827 − 0.996i)3-s + (−0.214 − 0.371i)5-s + (0.800 + 1.38i)7-s + (−0.986 − 0.164i)9-s + (0.802 + 1.39i)11-s + 1.90·13-s + (−0.387 + 0.182i)15-s + (−0.714 + 1.23i)17-s + (−0.626 − 0.779i)19-s + (1.44 − 0.682i)21-s + 0.740·23-s + (0.408 − 0.706i)25-s + (−0.245 + 0.969i)27-s + (−0.396 + 0.686i)29-s + (0.404 − 0.700i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.988 + 0.148i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69211 - 0.126154i\)
\(L(\frac12)\) \(\approx\) \(1.69211 - 0.126154i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.143 + 1.72i)T \)
19 \( 1 + (2.72 + 3.39i)T \)
good5 \( 1 + (0.479 + 0.830i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.11 - 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.66 - 4.61i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.85T + 13T^{2} \)
17 \( 1 + (2.94 - 5.10i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 3.55T + 23T^{2} \)
29 \( 1 + (2.13 - 3.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.25 + 3.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.49T + 37T^{2} \)
41 \( 1 + (5.97 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.0594T + 43T^{2} \)
47 \( 1 + (0.896 - 1.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.630 - 1.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.86 + 3.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.56 - 2.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (-4.83 + 8.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.40 - 5.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + (2.39 + 4.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−10.78556157062409007131929439442, −9.053401498303711353420704123167, −8.744649414688581825819659311073, −8.072497329225247187666795265610, −6.78081457866909360937779057871, −6.18847447274571616158443693099, −5.08945644611617898900441219212, −3.94330578810229684613276788355, −2.28464947512668472177893177190, −1.47217822525539321390909198019, 1.08275771638463753574362375416, 3.27595480633302101546457661851, 3.85905786574631974182819276054, 4.78915387283966523485872260769, 6.03548704117070201380167685625, 6.89934461558012752612631525158, 8.217289636553962075936853401621, 8.649783295810419715193386585577, 9.708419575031903474669535623634, 10.85779062883715930039609491807

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