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Properties

Label 2-684-228.119-c1-0-2
Degree $2$
Conductor $684$
Sign $-0.717 - 0.696i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.797 − 1.16i)2-s + (−0.726 + 1.86i)4-s + (−1.60 + 1.91i)5-s + (1.74 − 1.00i)7-s + (2.75 − 0.637i)8-s + (3.51 + 0.348i)10-s + (−2.81 + 4.86i)11-s + (−0.0876 − 0.497i)13-s + (−2.56 − 1.23i)14-s + (−2.94 − 2.70i)16-s + (−1.72 − 4.74i)17-s + (−2.18 − 3.76i)19-s + (−2.39 − 4.38i)20-s + (7.92 − 0.602i)22-s + (−5.56 + 4.67i)23-s + ⋯
L(s)  = 1  + (−0.564 − 0.825i)2-s + (−0.363 + 0.931i)4-s + (−0.718 + 0.856i)5-s + (0.658 − 0.380i)7-s + (0.974 − 0.225i)8-s + (1.11 + 0.110i)10-s + (−0.847 + 1.46i)11-s + (−0.0243 − 0.137i)13-s + (−0.685 − 0.329i)14-s + (−0.735 − 0.677i)16-s + (−0.419 − 1.15i)17-s + (−0.502 − 0.864i)19-s + (−0.536 − 0.980i)20-s + (1.69 − 0.128i)22-s + (−1.16 + 0.974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0719380 + 0.177222i\)
\(L(\frac12)\) \(\approx\) \(0.0719380 + 0.177222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.797 + 1.16i)T \)
3 \( 1 \)
19 \( 1 + (2.18 + 3.76i)T \)
good5 \( 1 + (1.60 - 1.91i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.74 + 1.00i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.81 - 4.86i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0876 + 0.497i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (1.72 + 4.74i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.56 - 4.67i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-2.71 + 7.44i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.16 - 2.40i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.161T + 37T^{2} \)
41 \( 1 + (8.58 + 1.51i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.94 - 5.89i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (5.46 + 1.98i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-5.91 - 7.04i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-5.08 + 1.84i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.11 - 4.29i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.35 - 9.21i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (9.51 + 7.98i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.87 + 10.6i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-3.87 - 0.682i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.85 - 4.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.19 + 1.44i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.82 - 0.662i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.74064941147897946656217835679, −10.13426928628904552634341662449, −9.302541850526940418870034115131, −8.058207951288492668928549716972, −7.51670799526838009905925160718, −6.88021319313904072149288328770, −4.95495324227076108791057571816, −4.21592736313132300819851293573, −2.98019506741670689774429348828, −1.96610961645431719034707389149, 0.11830619417736807717256437288, 1.78158560166382699050250773266, 3.80538318883952079900231701437, 4.88989415900520727258911924310, 5.67666173954223644506395387321, 6.59085639190225258059263474593, 7.987570629743578831004164781198, 8.437368640128895660804518642348, 8.671560812057590472546568745334, 10.19235576629184440173774850837

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