Properties

Label 2-684-3.2-c2-0-4
Degree $2$
Conductor $684$
Sign $0.577 - 0.816i$
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  − 1.18i·5-s + 11.5·7-s + 16.8i·11-s − 14.6·13-s + 23.3i·17-s + 4.35·19-s + 11.6i·23-s + 23.6·25-s − 43.4i·29-s + 7.00·31-s − 13.6i·35-s − 71.2·37-s + 5.74i·41-s + 78.3·43-s + 59.9i·47-s + ⋯
L(s)  = 1  − 0.236i·5-s + 1.65·7-s + 1.52i·11-s − 1.12·13-s + 1.37i·17-s + 0.229·19-s + 0.505i·23-s + 0.944·25-s − 1.49i·29-s + 0.225·31-s − 0.390i·35-s − 1.92·37-s + 0.140i·41-s + 1.82·43-s + 1.27i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.994301696\)
\(L(\frac12)\) \(\approx\) \(1.994301696\)
\(L(2)\) 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 \( 1 - 4.35T \)
good5 \( 1 + 1.18iT - 25T^{2} \)
7 \( 1 - 11.5T + 49T^{2} \)
11 \( 1 - 16.8iT - 121T^{2} \)
13 \( 1 + 14.6T + 169T^{2} \)
17 \( 1 - 23.3iT - 289T^{2} \)
23 \( 1 - 11.6iT - 529T^{2} \)
29 \( 1 + 43.4iT - 841T^{2} \)
31 \( 1 - 7.00T + 961T^{2} \)
37 \( 1 + 71.2T + 1.36e3T^{2} \)
41 \( 1 - 5.74iT - 1.68e3T^{2} \)
43 \( 1 - 78.3T + 1.84e3T^{2} \)
47 \( 1 - 59.9iT - 2.20e3T^{2} \)
53 \( 1 - 96.4iT - 2.80e3T^{2} \)
59 \( 1 - 20.9iT - 3.48e3T^{2} \)
61 \( 1 - 29.1T + 3.72e3T^{2} \)
67 \( 1 + 5.31T + 4.48e3T^{2} \)
71 \( 1 - 84.5iT - 5.04e3T^{2} \)
73 \( 1 - 61.3T + 5.32e3T^{2} \)
79 \( 1 - 16.8T + 6.24e3T^{2} \)
83 \( 1 + 123. iT - 6.88e3T^{2} \)
89 \( 1 - 129. iT - 7.92e3T^{2} \)
97 \( 1 - 116.T + 9.40e3T^{2} \)
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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

−10.42670322337897175328559869978, −9.595493382428019602933306831124, −8.596106097542695988946394494404, −7.72468123040116114647714298389, −7.18514783138408905067586483408, −5.74774333208567018452796843620, −4.76030636518859017918327363230, −4.25219912446975389848312209159, −2.36514199342682211766949383064, −1.44808807953220604856563984371, 0.76477416333816228785994789530, 2.27786050417900066606390341237, 3.42829716619328237113555244566, 4.99165052413698871443069399259, 5.23046168061602320561990673884, 6.77162308958691571274643938285, 7.53481688137220486458474737867, 8.475104150858923920627279860199, 9.067465247117845330308937187283, 10.38621560282585608921803395560

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