Properties

Label 2-684-228.119-c1-0-9
Degree $2$
Conductor $684$
Sign $-0.847 - 0.530i$
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  + (−0.268 + 1.38i)2-s + (−1.85 − 0.745i)4-s + (0.280 − 0.334i)5-s + (−1.12 + 0.646i)7-s + (1.53 − 2.37i)8-s + (0.389 + 0.479i)10-s + (0.842 − 1.45i)11-s + (0.944 + 5.35i)13-s + (−0.597 − 1.72i)14-s + (2.88 + 2.76i)16-s + (0.614 + 1.68i)17-s + (−3.19 + 2.96i)19-s + (−0.770 + 0.411i)20-s + (1.80 + 1.56i)22-s + (−1.92 + 1.61i)23-s + ⋯
L(s)  = 1  + (−0.189 + 0.981i)2-s + (−0.927 − 0.372i)4-s + (0.125 − 0.149i)5-s + (−0.423 + 0.244i)7-s + (0.541 − 0.840i)8-s + (0.123 + 0.151i)10-s + (0.254 − 0.439i)11-s + (0.261 + 1.48i)13-s + (−0.159 − 0.462i)14-s + (0.722 + 0.691i)16-s + (0.149 + 0.409i)17-s + (−0.733 + 0.680i)19-s + (−0.172 + 0.0920i)20-s + (0.383 + 0.332i)22-s + (−0.401 + 0.337i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.847 - 0.530i$
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: \((2,\ 684,\ (\ :1/2),\ -0.847 - 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258888 + 0.901302i\)
\(L(\frac12)\) \(\approx\) \(0.258888 + 0.901302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.268 - 1.38i)T \)
3 \( 1 \)
19 \( 1 + (3.19 - 2.96i)T \)
good5 \( 1 + (-0.280 + 0.334i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.12 - 0.646i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.842 + 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.944 - 5.35i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.614 - 1.68i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (1.92 - 1.61i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.0255 + 0.0702i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.87 - 1.66i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.65T + 37T^{2} \)
41 \( 1 + (6.88 + 1.21i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.06 - 3.65i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-10.3 - 3.76i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (2.66 + 3.17i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.983 - 0.357i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.05 + 0.889i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (4.42 - 12.1i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.20 - 2.68i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.72 + 9.78i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-3.47 - 0.612i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.00 + 6.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.01 + 0.355i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.95 + 1.80i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.65259783208783390560012641698, −9.631656855535684007684187588161, −9.039208250820206041804296914736, −8.296816735456854185283498432915, −7.24221237569561235842445182999, −6.35063627133078789716236428314, −5.75636989094623398691829697414, −4.50966830528751006492845161860, −3.58996575188174865480996265021, −1.60525017653250923387957942924, 0.55021876856309131387779260205, 2.26699144791559786447704615692, 3.28288281477525481891979077580, 4.33370574353556470339245064619, 5.40555251694558760291649594322, 6.59687337815481474769391941629, 7.73962836381501561430884098477, 8.551435924245651441194492061306, 9.491444847330310379556035118703, 10.29039714794948851241803623889

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