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Properties

Label 2-684-684.535-c1-0-85
Degree $2$
Conductor $684$
Sign $0.149 + 0.988i$
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  + (1.41 − 0.0331i)2-s + (−1.71 − 0.275i)3-s + (1.99 − 0.0937i)4-s + (−0.487 − 0.409i)5-s + (−2.42 − 0.332i)6-s + (−1.01 − 0.586i)7-s + (2.82 − 0.198i)8-s + (2.84 + 0.940i)9-s + (−0.702 − 0.562i)10-s − 3.53i·11-s + (−3.44 − 0.389i)12-s + (−3.69 − 4.39i)13-s + (−1.45 − 0.796i)14-s + (0.721 + 0.833i)15-s + (3.98 − 0.374i)16-s + (0.0453 + 0.0380i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0234i)2-s + (−0.987 − 0.158i)3-s + (0.998 − 0.0468i)4-s + (−0.218 − 0.182i)5-s + (−0.990 − 0.135i)6-s + (−0.384 − 0.221i)7-s + (0.997 − 0.0703i)8-s + (0.949 + 0.313i)9-s + (−0.222 − 0.177i)10-s − 1.06i·11-s + (−0.993 − 0.112i)12-s + (−1.02 − 1.21i)13-s + (−0.389 − 0.212i)14-s + (0.186 + 0.215i)15-s + (0.995 − 0.0936i)16-s + (0.0110 + 0.00923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.149 + 0.988i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.149 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35512 - 1.16515i\)
\(L(\frac12)\) \(\approx\) \(1.35512 - 1.16515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0331i)T \)
3 \( 1 + (1.71 + 0.275i)T \)
19 \( 1 + (-3.92 - 1.89i)T \)
good5 \( 1 + (0.487 + 0.409i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (1.01 + 0.586i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.53iT - 11T^{2} \)
13 \( 1 + (3.69 + 4.39i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.0453 - 0.0380i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (0.329 - 0.905i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-3.19 + 8.78i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 - 1.69T + 31T^{2} \)
37 \( 1 - 7.20iT - 37T^{2} \)
41 \( 1 + (-4.10 - 0.724i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (4.34 + 11.9i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (2.98 - 8.18i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.85 + 0.680i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (-6.13 + 2.23i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.36 - 6.17i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.22 - 6.93i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (1.72 - 9.80i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (9.78 - 3.56i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-3.96 - 3.33i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-8.82 - 5.09i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.40 + 3.84i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (7.75 + 1.36i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.24587287807836035811677557023, −10.07297224144616366908846453124, −8.167262628062793936291722765515, −7.46784625042315300092860328208, −6.41828118861686896086559019197, −5.70680090965325762990101196352, −4.94096300831706088762315663370, −3.85395746346698405911865555493, −2.68873250034056654017181743722, −0.78500712419351330858792439637, 1.81516874515867159914469625704, 3.26647721955884697570863225512, 4.55514048328656603068599619843, 4.99355471329213982796127286627, 6.15750944505569763064599960881, 7.05106433941084963625848126435, 7.40521035305041733297150075844, 9.291775671583149129589207789475, 9.971459770230316279759174091686, 10.96430948368427240151039449909

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