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Properties

Label 2-684-19.8-c2-0-13
Degree $2$
Conductor $684$
Sign $0.483 + 0.875i$
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  + (4.81 − 8.34i)5-s + 7.59·7-s + 4.61·11-s + (19.8 − 11.4i)13-s + (−3.02 + 5.23i)17-s + (−2.21 + 18.8i)19-s + (11.8 + 20.5i)23-s + (−33.9 − 58.7i)25-s + (−10.1 + 5.87i)29-s + 4.22i·31-s + (36.5 − 63.3i)35-s + 11.8i·37-s + (7.64 + 4.41i)41-s + (−11.5 + 19.9i)43-s + (−33.0 − 57.3i)47-s + ⋯
L(s)  = 1  + (0.963 − 1.66i)5-s + 1.08·7-s + 0.419·11-s + (1.52 − 0.880i)13-s + (−0.177 + 0.307i)17-s + (−0.116 + 0.993i)19-s + (0.515 + 0.893i)23-s + (−1.35 − 2.34i)25-s + (−0.351 + 0.202i)29-s + 0.136i·31-s + (1.04 − 1.80i)35-s + 0.320i·37-s + (0.186 + 0.107i)41-s + (−0.267 + 0.463i)43-s + (−0.704 − 1.21i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.483 + 0.875i$
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: \((2,\ 684,\ (\ :1),\ 0.483 + 0.875i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.686291357\)
\(L(\frac12)\) \(\approx\) \(2.686291357\)
\(L(2)\) 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 \( 1 + (2.21 - 18.8i)T \)
good5 \( 1 + (-4.81 + 8.34i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 7.59T + 49T^{2} \)
11 \( 1 - 4.61T + 121T^{2} \)
13 \( 1 + (-19.8 + 11.4i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (3.02 - 5.23i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-11.8 - 20.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (10.1 - 5.87i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 4.22iT - 961T^{2} \)
37 \( 1 - 11.8iT - 1.36e3T^{2} \)
41 \( 1 + (-7.64 - 4.41i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (11.5 - 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (33.0 + 57.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (40.7 - 23.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-7.52 - 4.34i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (3.57 + 6.19i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (77.3 - 44.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-67.9 - 39.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-43.1 + 74.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-57.9 - 33.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 5.30T + 6.88e3T^{2} \)
89 \( 1 + (-99.6 + 57.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (49.2 + 28.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.04194794694821688496344073396, −9.114805019078047438600705300276, −8.444240229787773200369228515163, −7.915473416152231106311351623218, −6.24250972474641417072257980702, −5.53426135803168120104964616877, −4.80094878067274905770758631092, −3.68508683543514630285801726472, −1.71514647229517825525324019363, −1.15802739425968110391081189227, 1.57155151910284839020517528730, 2.56391157098206016764141824839, 3.77006102934894768689115592087, 5.00737409007741728028268170692, 6.29657838892340513537290613454, 6.63160009597252051347089449240, 7.69065355611012775928467057218, 8.850501199749988211774216266333, 9.536838830479128314821770497557, 10.80245892884214182235450741760

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