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Properties

Label 2-684-171.103-c2-0-33
Degree $2$
Conductor $684$
Sign $-0.739 + 0.673i$
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  + (1.44 + 2.62i)3-s + (−1.78 − 3.09i)5-s + (2.11 + 3.67i)7-s + (−4.81 + 7.60i)9-s + (−8.61 − 14.9i)11-s − 19.3i·13-s + (5.55 − 9.17i)15-s + (−16.5 + 28.5i)17-s + (−0.306 + 18.9i)19-s + (−6.58 + 10.8i)21-s − 33.6·23-s + (6.11 − 10.5i)25-s + (−26.9 − 1.68i)27-s + (0.310 + 0.179i)29-s + (−43.6 − 25.1i)31-s + ⋯
L(s)  = 1  + (0.481 + 0.876i)3-s + (−0.357 − 0.618i)5-s + (0.302 + 0.524i)7-s + (−0.535 + 0.844i)9-s + (−0.782 − 1.35i)11-s − 1.49i·13-s + (0.370 − 0.611i)15-s + (−0.970 + 1.68i)17-s + (−0.0161 + 0.999i)19-s + (−0.313 + 0.517i)21-s − 1.46·23-s + (0.244 − 0.423i)25-s + (−0.998 − 0.0622i)27-s + (0.0106 + 0.00617i)29-s + (−1.40 − 0.812i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.739 + 0.673i$
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: \((2,\ 684,\ (\ :1),\ -0.739 + 0.673i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2348411208\)
\(L(\frac12)\) \(\approx\) \(0.2348411208\)
\(L(2)\) 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 \( 1 + (-1.44 - 2.62i)T \)
19 \( 1 + (0.306 - 18.9i)T \)
good5 \( 1 + (1.78 + 3.09i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.11 - 3.67i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (8.61 + 14.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 19.3iT - 169T^{2} \)
17 \( 1 + (16.5 - 28.5i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + 33.6T + 529T^{2} \)
29 \( 1 + (-0.310 - 0.179i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (43.6 + 25.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 26.4iT - 1.36e3T^{2} \)
41 \( 1 + (34.5 - 19.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 19.8T + 1.84e3T^{2} \)
47 \( 1 + (2.57 - 4.46i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-20.3 + 11.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (23.0 - 13.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.5 + 39.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 79.5iT - 4.48e3T^{2} \)
71 \( 1 + (1.79 + 1.03i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-18.0 + 31.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 71.7iT - 6.24e3T^{2} \)
83 \( 1 + (63.8 + 110. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (35.6 - 20.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 50.6iT - 9.40e3T^{2} \)
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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

−10.11569247613145775432236878278, −8.817095045816494212522675999786, −8.248074465906419614923031402296, −7.952550537418482891346415767812, −5.84265468105222421730361662285, −5.55119132303914760522283053030, −4.21365523447699932802129142667, −3.42968301700000616982308039023, −2.14482789671016753653776726416, −0.07193829545051049564586048781, 1.83227192403506643385857603722, 2.69012409164119473651415788389, 4.08076898234568261459457715061, 5.03589880203007044696401914913, 6.73604727644630475430333208638, 7.08355787059144017298224455330, 7.67636683028676196430652352654, 8.925020466560038409135136742185, 9.559433837138592199307224553660, 10.73791264566568081618200039257

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