Properties

Label 2-684-19.12-c2-0-4
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: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.0186 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.876253348\)
\(L(\frac12)\) \(\approx\) \(1.876253348\)
\(L(2)\) 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

−10.28806634521620971294769502062, −9.863114120911182500633052158483, −8.921341043391507917980520536933, −7.78395284027341119509683445826, −6.62431723279747100663719450413, −6.39882119277928507687570766503, −5.17479331099076940012974939984, −3.69620568110846372210615494531, −2.89741487280811206721813905389, −1.53717491798908231660460952428, 0.69847906787698184686371475923, 1.94074133881091418927663379065, 3.46190443158437838916187159338, 4.63757081478656568777585039103, 5.50055238151727439826115812846, 6.39271177830041892676827018234, 7.40625146102387951796917031690, 8.577533415172292310351098969181, 9.360466500853280667971840880758, 9.654545451693880276263556129629

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