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Properties

Label 2-684-4.3-c2-0-11
Degree $2$
Conductor $684$
Sign $-0.980 + 0.194i$
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  + (−1.26 + 1.54i)2-s + (−0.779 − 3.92i)4-s + 6.46·5-s + 11.6i·7-s + (7.05 + 3.77i)8-s + (−8.20 + 9.99i)10-s − 8.68i·11-s − 21.6·13-s + (−18.0 − 14.7i)14-s + (−14.7 + 6.11i)16-s − 26.7·17-s + 4.35i·19-s + (−5.03 − 25.3i)20-s + (13.4 + 11.0i)22-s + 15.9i·23-s + ⋯
L(s)  = 1  + (−0.634 + 0.772i)2-s + (−0.194 − 0.980i)4-s + 1.29·5-s + 1.66i·7-s + (0.881 + 0.471i)8-s + (−0.820 + 0.999i)10-s − 0.789i·11-s − 1.66·13-s + (−1.28 − 1.05i)14-s + (−0.924 + 0.382i)16-s − 1.57·17-s + 0.229i·19-s + (−0.251 − 1.26i)20-s + (0.610 + 0.500i)22-s + 0.693i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.980 + 0.194i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.980 + 0.194i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6851806968\)
\(L(\frac12)\) \(\approx\) \(0.6851806968\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 1.54i)T \)
3 \( 1 \)
19 \( 1 - 4.35iT \)
good5 \( 1 - 6.46T + 25T^{2} \)
7 \( 1 - 11.6iT - 49T^{2} \)
11 \( 1 + 8.68iT - 121T^{2} \)
13 \( 1 + 21.6T + 169T^{2} \)
17 \( 1 + 26.7T + 289T^{2} \)
23 \( 1 - 15.9iT - 529T^{2} \)
29 \( 1 - 27.3T + 841T^{2} \)
31 \( 1 - 37.0iT - 961T^{2} \)
37 \( 1 + 38.5T + 1.36e3T^{2} \)
41 \( 1 + 18.1T + 1.68e3T^{2} \)
43 \( 1 + 30.1iT - 1.84e3T^{2} \)
47 \( 1 - 2.86iT - 2.20e3T^{2} \)
53 \( 1 + 35.6T + 2.80e3T^{2} \)
59 \( 1 + 4.73iT - 3.48e3T^{2} \)
61 \( 1 - 75.8T + 3.72e3T^{2} \)
67 \( 1 - 105. iT - 4.48e3T^{2} \)
71 \( 1 - 20.2iT - 5.04e3T^{2} \)
73 \( 1 + 27.7T + 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 + 33.6iT - 6.88e3T^{2} \)
89 \( 1 + 72.7T + 7.92e3T^{2} \)
97 \( 1 + 164.T + 9.40e3T^{2} \)
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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

−10.34274617519310124787877542546, −9.681024990375787201490830134194, −8.892717682321126853086215114171, −8.498889957200482063167303014024, −7.07236703057290404878382132845, −6.28795100208903234221115077771, −5.47644342016555114226790286138, −4.94144127100360433915025501901, −2.63421071769089225096030471952, −1.84473229512227667456454614359, 0.27512076782534417971952440321, 1.79429754775107871423978599001, 2.63288681806544275005345584804, 4.24932564863075020778639983375, 4.85215158347943847812317345888, 6.69022858024773383913053871263, 7.11719001445902060437581819357, 8.182731799633082633513620555905, 9.416791478324378944336911420547, 9.869246905503225680482612924120

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