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Properties

Label 2-684-19.8-c2-0-0
Degree $2$
Conductor $684$
Sign $-0.921 + 0.388i$
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  + (−2.50 + 4.34i)5-s + 7.91·7-s − 16.3·11-s + (−11.1 + 6.43i)13-s + (4.46 − 7.73i)17-s + (−18.5 − 4.20i)19-s + (−7.27 − 12.6i)23-s + (−0.0975 − 0.168i)25-s + (29.7 − 17.1i)29-s + 25.5i·31-s + (−19.8 + 34.4i)35-s − 49.2i·37-s + (−27.8 − 16.0i)41-s + (−4.51 + 7.81i)43-s + (−19.9 − 34.5i)47-s + ⋯
L(s)  = 1  + (−0.501 + 0.869i)5-s + 1.13·7-s − 1.48·11-s + (−0.857 + 0.495i)13-s + (0.262 − 0.455i)17-s + (−0.975 − 0.221i)19-s + (−0.316 − 0.548i)23-s + (−0.00390 − 0.00675i)25-s + (1.02 − 0.592i)29-s + 0.823i·31-s + (−0.567 + 0.983i)35-s − 1.33i·37-s + (−0.678 − 0.391i)41-s + (−0.104 + 0.181i)43-s + (−0.424 − 0.734i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.921 + 0.388i$
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: \((2,\ 684,\ (\ :1),\ -0.921 + 0.388i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.04715583628\)
\(L(\frac12)\) \(\approx\) \(0.04715583628\)
\(L(2)\) 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 \( 1 + (18.5 + 4.20i)T \)
good5 \( 1 + (2.50 - 4.34i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 7.91T + 49T^{2} \)
11 \( 1 + 16.3T + 121T^{2} \)
13 \( 1 + (11.1 - 6.43i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-4.46 + 7.73i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (7.27 + 12.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-29.7 + 17.1i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 25.5iT - 961T^{2} \)
37 \( 1 + 49.2iT - 1.36e3T^{2} \)
41 \( 1 + (27.8 + 16.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (4.51 - 7.81i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (19.9 + 34.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (73.6 - 42.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (53.9 + 31.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (38.1 + 66.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (69.8 - 40.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-11.0 - 6.37i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (42.2 - 73.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (106. + 61.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 21.9T + 6.88e3T^{2} \)
89 \( 1 + (-98.0 + 56.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-146. - 84.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.68408357831635781864837068573, −10.24416711856514051850974120917, −8.915358678546615510863567546981, −7.943405101230829097888968115011, −7.46133569092836249034585789402, −6.47039209964549698220959460934, −5.13094189670109352634917998049, −4.49206924025103980242825779612, −3.01729640888822250729789702503, −2.08176004261845639375842515134, 0.01595219982601826374210418592, 1.60907196364124311943985958729, 2.96554909349977747170315550081, 4.61450191391323488578593091120, 4.88500610875259322596356229971, 6.03119908964820347312647932152, 7.54942173329740767755816675910, 8.083093147559372157467198205407, 8.596940440970461724598442184857, 9.958884056203416216053494465094

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