Properties

Label 2-684-19.8-c2-0-9
Degree $2$
Conductor $684$
Sign $-0.0186 + 0.999i$
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  + (−3.48 + 6.04i)5-s − 3.44·7-s − 10.1·11-s + (−0.151 + 0.0874i)13-s + (−1.56 + 2.71i)17-s + (11.3 − 15.2i)19-s + (−3.48 − 6.04i)23-s + (−11.8 − 20.5i)25-s + (4.70 − 2.71i)29-s − 36.8i·31-s + (12.0 − 20.8i)35-s − 27.1i·37-s + (51.2 + 29.6i)41-s + (10.9 − 19.0i)43-s + (−6.27 − 10.8i)47-s + ⋯
L(s)  = 1  + (−0.697 + 1.20i)5-s − 0.492·7-s − 0.919·11-s + (−0.0116 + 0.00672i)13-s + (−0.0922 + 0.159i)17-s + (0.597 − 0.802i)19-s + (−0.151 − 0.262i)23-s + (−0.473 − 0.820i)25-s + (0.162 − 0.0936i)29-s − 1.18i·31-s + (0.343 − 0.595i)35-s − 0.734i·37-s + (1.25 + 0.722i)41-s + (0.255 − 0.441i)43-s + (−0.133 − 0.231i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0186 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0186 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.0186 + 0.999i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.0186 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5600540579\)
\(L(\frac12)\) \(\approx\) \(0.5600540579\)
\(L(2)\) 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 \( 1 \)
19 \( 1 + (-11.3 + 15.2i)T \)
good5 \( 1 + (3.48 - 6.04i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 3.44T + 49T^{2} \)
11 \( 1 + 10.1T + 121T^{2} \)
13 \( 1 + (0.151 - 0.0874i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (1.56 - 2.71i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (3.48 + 6.04i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-4.70 + 2.71i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 36.8iT - 961T^{2} \)
37 \( 1 + 27.1iT - 1.36e3T^{2} \)
41 \( 1 + (-51.2 - 29.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-10.9 + 19.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (6.27 + 10.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-36.1 + 20.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (21.9 + 12.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (26.1 + 45.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-41.3 + 23.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (14.1 + 8.14i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-14.9 + 25.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (53.1 + 30.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 148.T + 6.88e3T^{2} \)
89 \( 1 + (54.9 - 31.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-27 - 15.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.15826710637087075010759512983, −9.336696346058216599316006247379, −8.097917135474524708562726484997, −7.41664770961978467702495030814, −6.64653507822724800185144258970, −5.65650966285808836936936276072, −4.35846468468732824129422781917, −3.25864949806396695890905100104, −2.46806691719167902033152938229, −0.22086700658372112064774587270, 1.18044273800897214490308390394, 2.90266625916954492714415457786, 4.06829966728666885950444712568, 5.01299272625270131962170584896, 5.83907976046492246391840475376, 7.16674621441820068547003344339, 7.991985053628139319581140135392, 8.682048666677123135147138859281, 9.588792924562175738895999665776, 10.41555889547680125963695836038

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