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Properties

Label 2-684-19.9-c1-0-0
Degree $2$
Conductor $684$
Sign $-0.585 - 0.810i$
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  + (−3.00 − 1.09i)5-s + (0.278 − 0.482i)7-s + (1.96 + 3.39i)11-s + (−3.19 − 2.67i)13-s + (0.660 + 3.74i)17-s + (−1.84 + 3.94i)19-s + (−4.67 + 1.70i)23-s + (4.01 + 3.36i)25-s + (−0.0201 + 0.114i)29-s + (−3.54 + 6.13i)31-s + (−1.36 + 1.14i)35-s − 5.67·37-s + (−9.20 + 7.72i)41-s + (6.74 + 2.45i)43-s + (0.00419 − 0.0237i)47-s + ⋯
L(s)  = 1  + (−1.34 − 0.489i)5-s + (0.105 − 0.182i)7-s + (0.591 + 1.02i)11-s + (−0.885 − 0.743i)13-s + (0.160 + 0.907i)17-s + (−0.423 + 0.906i)19-s + (−0.974 + 0.354i)23-s + (0.803 + 0.673i)25-s + (−0.00374 + 0.0212i)29-s + (−0.636 + 1.10i)31-s + (−0.230 + 0.193i)35-s − 0.933·37-s + (−1.43 + 1.20i)41-s + (1.02 + 0.374i)43-s + (0.000612 − 0.00347i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.585 - 0.810i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206322 + 0.403762i\)
\(L(\frac12)\) \(\approx\) \(0.206322 + 0.403762i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (1.84 - 3.94i)T \)
good5 \( 1 + (3.00 + 1.09i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.278 + 0.482i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.96 - 3.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.19 + 2.67i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.660 - 3.74i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (4.67 - 1.70i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0201 - 0.114i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.54 - 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.67T + 37T^{2} \)
41 \( 1 + (9.20 - 7.72i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-6.74 - 2.45i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.00419 + 0.0237i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-8.18 + 2.97i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.88 + 10.7i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (11.7 - 4.29i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.27 + 12.8i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (3.17 + 1.15i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-0.338 + 0.284i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-1.31 + 1.10i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.90 + 5.02i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.83 + 1.53i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-1.86 - 10.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.69051243328138118853934182806, −10.03272699336815736393981970651, −8.957559764197055195077027619904, −7.962028854984609986710197949954, −7.58944175068088696256140962812, −6.45966465624916785078288649779, −5.14383210913896890810042930546, −4.25434241334127264912566593020, −3.47032189696666292183787991762, −1.67273579544996517805418705824, 0.23748217301132697292252776453, 2.43110238799364294613753672715, 3.64103222545964653123051351380, 4.42157966261155653113156495672, 5.67353852603185628243502771145, 6.92797338486160916311672881880, 7.38906280409713477945379525687, 8.503958270405298982897941926952, 9.132317570654748884063455766517, 10.31567665212327092243985449431

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