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Properties

Label 2-684-171.103-c2-0-28
Degree $2$
Conductor $684$
Sign $-0.997 + 0.0672i$
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  + (−1.91 − 2.31i)3-s + (−2.98 − 5.16i)5-s + (6.83 + 11.8i)7-s + (−1.67 + 8.84i)9-s + (−0.0305 − 0.0528i)11-s − 20.6i·13-s + (−6.22 + 16.7i)15-s + (15.1 − 26.1i)17-s + (−16.6 + 9.20i)19-s + (14.2 − 38.4i)21-s − 1.71·23-s + (−5.27 + 9.13i)25-s + (23.6 − 13.0i)27-s + (−44.1 − 25.4i)29-s + (2.97 + 1.71i)31-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)3-s + (−0.596 − 1.03i)5-s + (0.976 + 1.69i)7-s + (−0.186 + 0.982i)9-s + (−0.00277 − 0.00480i)11-s − 1.58i·13-s + (−0.414 + 1.11i)15-s + (0.888 − 1.53i)17-s + (−0.874 + 0.484i)19-s + (0.679 − 1.83i)21-s − 0.0743·23-s + (−0.210 + 0.365i)25-s + (0.875 − 0.483i)27-s + (−1.52 − 0.878i)29-s + (0.0960 + 0.0554i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0672i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.997 + 0.0672i$
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: \((2,\ 684,\ (\ :1),\ -0.997 + 0.0672i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6260854224\)
\(L(\frac12)\) \(\approx\) \(0.6260854224\)
\(L(2)\) 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 \( 1 + (1.91 + 2.31i)T \)
19 \( 1 + (16.6 - 9.20i)T \)
good5 \( 1 + (2.98 + 5.16i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.83 - 11.8i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.0305 + 0.0528i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 20.6iT - 169T^{2} \)
17 \( 1 + (-15.1 + 26.1i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 1.71T + 529T^{2} \)
29 \( 1 + (44.1 + 25.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-2.97 - 1.71i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 18.8iT - 1.36e3T^{2} \)
41 \( 1 + (37.4 - 21.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 41.1T + 1.84e3T^{2} \)
47 \( 1 + (-4.06 + 7.03i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.9 + 22.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (40.2 - 23.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (38.4 - 66.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 41.1iT - 4.48e3T^{2} \)
71 \( 1 + (35.0 + 20.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-19.3 + 33.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 25.0iT - 6.24e3T^{2} \)
83 \( 1 + (-17.9 - 31.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (35.8 - 20.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−9.817678600629290783747022922855, −8.621964660379206247555223132549, −8.157513478066843909617702751200, −7.49362405006680299677370549357, −5.92565684595049119448244815070, −5.37151682350980299426782658571, −4.72658412758257142818836630957, −2.85737871522155850426341434236, −1.60633590433865885870337135434, −0.25061738182746121514388494890, 1.57714730926913542131825966637, 3.75946285503026041878198226327, 3.94744789879467679770535985898, 5.05781935385531320871782291912, 6.47437422735113617416742046804, 7.06025431355980492912271463289, 7.945203820236097824238018784891, 9.082713079791676159472446286519, 10.28944642957755236192515439353, 10.71832108048393001916787671281

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