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Properties

Label 2-684-171.103-c2-0-2
Degree $2$
Conductor $684$
Sign $-0.561 - 0.827i$
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  + (−2.96 − 0.480i)3-s + (−0.920 − 1.59i)5-s + (3.31 + 5.74i)7-s + (8.53 + 2.84i)9-s + (−7.65 − 13.2i)11-s + 2.53i·13-s + (1.95 + 5.16i)15-s + (−12.9 + 22.4i)17-s + (7.42 − 17.4i)19-s + (−7.05 − 18.5i)21-s + 5.08·23-s + (10.8 − 18.7i)25-s + (−23.9 − 12.5i)27-s + (−12.8 − 7.43i)29-s + (34.8 + 20.1i)31-s + ⋯
L(s)  = 1  + (−0.987 − 0.160i)3-s + (−0.184 − 0.318i)5-s + (0.473 + 0.820i)7-s + (0.948 + 0.316i)9-s + (−0.696 − 1.20i)11-s + 0.194i·13-s + (0.130 + 0.344i)15-s + (−0.761 + 1.31i)17-s + (0.390 − 0.920i)19-s + (−0.335 − 0.885i)21-s + 0.221·23-s + (0.432 − 0.748i)25-s + (−0.885 − 0.464i)27-s + (−0.444 − 0.256i)29-s + (1.12 + 0.649i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.561 - 0.827i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ -0.561 - 0.827i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4641840763\)
\(L(\frac12)\) \(\approx\) \(0.4641840763\)
\(L(2)\) 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 \( 1 + (2.96 + 0.480i)T \)
19 \( 1 + (-7.42 + 17.4i)T \)
good5 \( 1 + (0.920 + 1.59i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-3.31 - 5.74i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (7.65 + 13.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 2.53iT - 169T^{2} \)
17 \( 1 + (12.9 - 22.4i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 5.08T + 529T^{2} \)
29 \( 1 + (12.8 + 7.43i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-34.8 - 20.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 17.0iT - 1.36e3T^{2} \)
41 \( 1 + (46.5 - 26.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 50.4T + 1.84e3T^{2} \)
47 \( 1 + (32.3 - 55.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (88.6 - 51.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-47.8 + 27.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (41.5 - 72.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 - 111. iT - 4.48e3T^{2} \)
71 \( 1 + (-83.8 - 48.3i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (66.6 - 115. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 33.4iT - 6.24e3T^{2} \)
83 \( 1 + (45.0 + 78.0i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (26.4 - 15.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 79.5iT - 9.40e3T^{2} \)
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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

−10.82271470049048508150888699038, −9.867870777823733639773158274652, −8.537232599429926220024226681813, −8.251759407644741990746339130475, −6.82912944824118343928018775460, −6.04872450399118802307641031750, −5.19190330107193345273491882316, −4.41367745713452308445900870537, −2.81345778661968086926722516359, −1.31997165880294066217240780940, 0.20051993554854358555274109565, 1.78388094593516209283972375035, 3.47756575439132873262600793678, 4.73160622312676459721012962257, 5.14009840306944786434075687706, 6.58396127293015350953631659765, 7.24060437201540040268130088597, 7.932332551258650917740386930502, 9.464896413646355251218100805559, 10.12244868288821081516189753668

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