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Properties

Label 2-684-171.140-c2-0-7
Degree $2$
Conductor $684$
Sign $-0.965 - 0.259i$
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  + (−2.88 − 0.833i)3-s + 7.26i·5-s + (3.75 + 6.50i)7-s + (7.61 + 4.80i)9-s + (−14.6 + 8.48i)11-s + (10.1 + 17.5i)13-s + (6.05 − 20.9i)15-s + (12.9 − 7.46i)17-s + (−17.9 + 6.31i)19-s + (−5.40 − 21.8i)21-s + (28.8 − 16.6i)23-s − 27.8·25-s + (−17.9 − 20.1i)27-s − 30.3i·29-s + (−23.8 + 41.3i)31-s + ⋯
L(s)  = 1  + (−0.960 − 0.277i)3-s + 1.45i·5-s + (0.536 + 0.929i)7-s + (0.845 + 0.533i)9-s + (−1.33 + 0.771i)11-s + (0.780 + 1.35i)13-s + (0.403 − 1.39i)15-s + (0.760 − 0.439i)17-s + (−0.943 + 0.332i)19-s + (−0.257 − 1.04i)21-s + (1.25 − 0.723i)23-s − 1.11·25-s + (−0.664 − 0.747i)27-s − 1.04i·29-s + (−0.769 + 1.33i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.965 - 0.259i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.965 - 0.259i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9785171621\)
\(L(\frac12)\) \(\approx\) \(0.9785171621\)
\(L(2)\) 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 \( 1 + (2.88 + 0.833i)T \)
19 \( 1 + (17.9 - 6.31i)T \)
good5 \( 1 - 7.26iT - 25T^{2} \)
7 \( 1 + (-3.75 - 6.50i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (14.6 - 8.48i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.1 - 17.5i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.9 + 7.46i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (-28.8 + 16.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 30.3iT - 841T^{2} \)
31 \( 1 + (23.8 - 41.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 25.1T + 1.36e3T^{2} \)
41 \( 1 - 5.04iT - 1.68e3T^{2} \)
43 \( 1 + (24.8 - 43.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 62.4iT - 2.20e3T^{2} \)
53 \( 1 + (-47.5 - 27.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 73.6iT - 3.48e3T^{2} \)
61 \( 1 + 95.3T + 3.72e3T^{2} \)
67 \( 1 + (-0.588 - 1.01i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (12.3 - 7.13i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (59.5 + 103. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-0.0750 + 0.130i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (41.9 - 24.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-67.4 - 38.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (47.2 - 81.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.67905175564444608714427385329, −10.23332758123913136054534144688, −8.966950469877293946898930138124, −7.81131314428980105921387784310, −7.00854057098121036118465501102, −6.31316284324455677269256588274, −5.37824598409918825926588200113, −4.41422485472577860713706566466, −2.79686171061327977220382574770, −1.83910312819815202251684930859, 0.44384189611553870991993115575, 1.23285538446620382723740052548, 3.45567608382087021554163670204, 4.56667865908898452736253910883, 5.35362563033088255589540871747, 5.85953145297554020944741047713, 7.41956792802319230211821501140, 8.126228041276737473931667576692, 8.960960830873348322079466782917, 10.14321559302665434227564880642

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