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Properties

Label 2-684-19.12-c2-0-1
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  + (3 + 5.19i)5-s − 5·7-s + (−16.5 − 9.52i)13-s + (3 + 5.19i)17-s + (−13 + 13.8i)19-s + (−12 + 20.7i)23-s + (−5.5 + 9.52i)25-s + (−27 − 15.5i)29-s − 29.4i·31-s + (−15 − 25.9i)35-s − 60.6i·37-s + (36 − 20.7i)41-s + (12.5 + 21.6i)43-s + (−21 + 36.3i)47-s − 24·49-s + ⋯
L(s)  = 1  + (0.600 + 1.03i)5-s − 0.714·7-s + (−1.26 − 0.732i)13-s + (0.176 + 0.305i)17-s + (−0.684 + 0.729i)19-s + (−0.521 + 0.903i)23-s + (−0.220 + 0.381i)25-s + (−0.931 − 0.537i)29-s − 0.949i·31-s + (−0.428 − 0.742i)35-s − 1.63i·37-s + (0.878 − 0.506i)41-s + (0.290 + 0.503i)43-s + (−0.446 + 0.773i)47-s − 0.489·49-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: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.980 + 0.194i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2542508179\)
\(L(\frac12)\) \(\approx\) \(0.2542508179\)
\(L(2)\) 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 \( 1 + (13 - 13.8i)T \)
good5 \( 1 + (-3 - 5.19i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 5T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + (16.5 + 9.52i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (12 - 20.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (27 + 15.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 29.4iT - 961T^{2} \)
37 \( 1 + 60.6iT - 1.36e3T^{2} \)
41 \( 1 + (-36 + 20.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-12.5 - 21.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (21 - 36.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (54 + 31.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (63 - 36.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (21.5 - 37.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-49.5 - 28.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (54 - 31.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 126T + 6.88e3T^{2} \)
89 \( 1 + (-9 - 5.19i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (114 - 65.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.56647384704009064982594692848, −9.881624878995777175377063863104, −9.367029836696207122995430480782, −7.88233313358470050031356328436, −7.28543250336424490121613075555, −6.15051146277458375813302244145, −5.67325995674424755030461103674, −4.11929292337683045969788036633, −3.00913667717193496210934945475, −2.08630420622431793551764499965, 0.082841613900554177401101565810, 1.71518724832366238203402832107, 2.94795065271375083421373374208, 4.48749437261910730339440740044, 5.09495717838486606591370473522, 6.28374200190273088852360084825, 7.04105015892287891270121439627, 8.233802395712900664747308924230, 9.197994054782095037874489469637, 9.579117491332307959902552706799

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