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Properties

Label 2-684-19.12-c2-0-7
Degree $2$
Conductor $684$
Sign $0.980 - 0.194i$
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  + 13·7-s + (10.5 + 6.06i)13-s + (−13 + 13.8i)19-s + (12.5 − 21.6i)25-s − 19.0i·31-s + 12.1i·37-s + (30.5 + 52.8i)43-s + 120·49-s + (23.5 − 40.7i)61-s + (67.5 + 38.9i)67-s + (48.5 + 84.0i)73-s + (136.5 − 78.8i)79-s + (136.5 + 78.8i)91-s + (−168 + 96.9i)97-s − 202. i·103-s + ⋯
L(s)  = 1  + 1.85·7-s + (0.807 + 0.466i)13-s + (−0.684 + 0.729i)19-s + (0.5 − 0.866i)25-s − 0.614i·31-s + 0.327i·37-s + (0.709 + 1.22i)43-s + 2.44·49-s + (0.385 − 0.667i)61-s + (1.00 + 0.581i)67-s + (0.664 + 1.15i)73-s + (1.72 − 0.997i)79-s + (1.50 + 0.866i)91-s + (−1.73 + 0.999i)97-s − 1.96i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.980 - 0.194i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.980 - 0.194i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.351711424\)
\(L(\frac12)\) \(\approx\) \(2.351711424\)
\(L(2)\) 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 \( 1 \)
19 \( 1 + (13 - 13.8i)T \)
good5 \( 1 + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 13T + 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 + (-10.5 - 6.06i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + 19.0iT - 961T^{2} \)
37 \( 1 - 12.1iT - 1.36e3T^{2} \)
41 \( 1 + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.5 - 52.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.5 + 40.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-67.5 - 38.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-48.5 - 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-136.5 + 78.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (168 - 96.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.54358768100655405479735228280, −9.360061735854258310338517726280, −8.276082463720398058909713836093, −8.052667868568605460553743265002, −6.76513554613267130943483525054, −5.75484144356269236561634956255, −4.71726383279150290426044323114, −3.96969964801403487880992385640, −2.28147349569077036655308200473, −1.22212004018126805399558677477, 1.08326820611587270721232977971, 2.25830500374907296730574507924, 3.77537113244455535014350516935, 4.84606648182756085145170989563, 5.53699222615349270221559657762, 6.80984983746197292064188980193, 7.77545855470915191479054782441, 8.484347737596263028271913315572, 9.168182915854704806706613366894, 10.66902526474453197426225392510

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