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Properties

Label 2-684-19.8-c2-0-6
Degree $2$
Conductor $684$
Sign $0.813 - 0.582i$
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 + 1.73i)5-s − 7-s − 16·11-s + (13.5 − 7.79i)13-s + (11 − 19.0i)17-s + 19·19-s + (20 + 34.6i)23-s + (10.5 + 18.1i)25-s + (−15 + 8.66i)29-s + 50.2i·31-s + (1 − 1.73i)35-s − 15.5i·37-s + (12 + 6.92i)41-s + (24.5 − 42.4i)43-s + (23 + 39.8i)47-s + ⋯
L(s)  = 1  + (−0.200 + 0.346i)5-s − 0.142·7-s − 1.45·11-s + (1.03 − 0.599i)13-s + (0.647 − 1.12i)17-s + 19-s + (0.869 + 1.50i)23-s + (0.419 + 0.727i)25-s + (−0.517 + 0.298i)29-s + 1.62i·31-s + (0.0285 − 0.0494i)35-s − 0.421i·37-s + (0.292 + 0.168i)41-s + (0.569 − 0.986i)43-s + (0.489 + 0.847i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.813 - 0.582i$
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: \((2,\ 684,\ (\ :1),\ 0.813 - 0.582i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.629171908\)
\(L(\frac12)\) \(\approx\) \(1.629171908\)
\(L(2)\) 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 \( 1 - 19T \)
good5 \( 1 + (1 - 1.73i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + T + 49T^{2} \)
11 \( 1 + 16T + 121T^{2} \)
13 \( 1 + (-13.5 + 7.79i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-11 + 19.0i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-20 - 34.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (15 - 8.66i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 50.2iT - 961T^{2} \)
37 \( 1 + 15.5iT - 1.36e3T^{2} \)
41 \( 1 + (-12 - 6.92i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-24.5 + 42.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-23 - 39.8i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-42 + 24.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-57 - 32.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-48.5 - 84.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.5 - 12.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-42 - 24.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (17.5 - 30.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (76.5 + 44.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 146T + 6.88e3T^{2} \)
89 \( 1 + (-33 + 19.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-54 - 31.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.47581565875536462198855790007, −9.538194934722575689320531135803, −8.643813702587868256826476155281, −7.51307232752556592056700426166, −7.17540988824614920469543831922, −5.55687276631341101514144957769, −5.22792365114790030580674511415, −3.48602316821814774281724033308, −2.88516715063849563599291067144, −1.04997781685056742268644035198, 0.74717614808431635741248711236, 2.36268518168293775311217291288, 3.60824068578796079756010863262, 4.68046421249714392658196457289, 5.68843933235621825895167758185, 6.56661383186208885763347022746, 7.81859307845830652290363295572, 8.314700291319893249103639520274, 9.312121052167459808469163432601, 10.29008753756581841499783632647

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