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Properties

Label 2-684-76.31-c1-0-32
Degree $2$
Conductor $684$
Sign $0.189 + 0.981i$
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  + (−0.327 + 1.37i)2-s + (−1.78 − 0.902i)4-s + (0.139 + 0.242i)5-s − 1.55i·7-s + (1.82 − 2.15i)8-s + (−0.379 + 0.113i)10-s + 2.44i·11-s + (−5.47 − 3.16i)13-s + (2.13 + 0.509i)14-s + (2.37 + 3.22i)16-s + (−2.18 − 3.78i)17-s + (−3.17 + 2.99i)19-s + (−0.0311 − 0.559i)20-s + (−3.35 − 0.800i)22-s + (−5.71 − 3.30i)23-s + ⋯
L(s)  = 1  + (−0.231 + 0.972i)2-s + (−0.892 − 0.451i)4-s + (0.0625 + 0.108i)5-s − 0.586i·7-s + (0.645 − 0.763i)8-s + (−0.119 + 0.0357i)10-s + 0.735i·11-s + (−1.51 − 0.876i)13-s + (0.570 + 0.136i)14-s + (0.593 + 0.805i)16-s + (−0.529 − 0.917i)17-s + (−0.727 + 0.686i)19-s + (−0.00696 − 0.125i)20-s + (−0.715 − 0.170i)22-s + (−1.19 − 0.688i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363547 - 0.300213i\)
\(L(\frac12)\) \(\approx\) \(0.363547 - 0.300213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.327 - 1.37i)T \)
3 \( 1 \)
19 \( 1 + (3.17 - 2.99i)T \)
good5 \( 1 + (-0.139 - 0.242i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 1.55iT - 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (5.47 + 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.71 + 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.695 + 0.401i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 + 4.74iT - 37T^{2} \)
41 \( 1 + (-5.84 + 3.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.146 + 0.0845i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.19 + 0.691i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.25 + 2.45i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.05 + 1.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.58 - 2.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.00 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.74 - 3.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.43 + 9.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.00 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.99iT - 83T^{2} \)
89 \( 1 + (7.76 + 4.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.77 - 1.02i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.11189973918842777587209573420, −9.480436504213520158287195642154, −8.352455576848136460040719948020, −7.53187125569285778235716939332, −6.95815448747198823003502628855, −5.91285846582416559256939200297, −4.84649425953753914164215920821, −4.11849205902525760200777558955, −2.34578558450559381530278782416, −0.25824675827533346697424591306, 1.79514774817608033237029982030, 2.77872445988905871676215679997, 4.07303615094779227065278109711, 4.97502425140140056606628663230, 6.09768728522580155272826825185, 7.36841585667420219079443702884, 8.384724372415372161951365328371, 9.118325710347790564668399017485, 9.752189632533888906508323979622, 10.77349013091786089054643949056

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