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Properties

Label 2-684-76.51-c1-0-40
Degree $2$
Conductor $684$
Sign $-0.168 + 0.985i$
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  + (1.19 − 0.750i)2-s + (0.874 − 1.79i)4-s + (0.165 + 0.936i)5-s + (−2.67 − 1.54i)7-s + (−0.301 − 2.81i)8-s + (0.900 + 0.998i)10-s + (3.39 − 1.95i)11-s + (−0.284 − 0.781i)13-s + (−4.36 + 0.155i)14-s + (−2.47 − 3.14i)16-s + (3.14 − 2.63i)17-s + (−0.473 − 4.33i)19-s + (1.82 + 0.521i)20-s + (2.59 − 4.89i)22-s + (−2.45 − 0.432i)23-s + ⋯
L(s)  = 1  + (0.847 − 0.530i)2-s + (0.437 − 0.899i)4-s + (0.0738 + 0.418i)5-s + (−1.01 − 0.583i)7-s + (−0.106 − 0.994i)8-s + (0.284 + 0.315i)10-s + (1.02 − 0.590i)11-s + (−0.0789 − 0.216i)13-s + (−1.16 + 0.0415i)14-s + (−0.617 − 0.786i)16-s + (0.762 − 0.639i)17-s + (−0.108 − 0.994i)19-s + (0.408 + 0.116i)20-s + (0.553 − 1.04i)22-s + (−0.511 − 0.0901i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44131 - 1.70908i\)
\(L(\frac12)\) \(\approx\) \(1.44131 - 1.70908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.750i)T \)
3 \( 1 \)
19 \( 1 + (0.473 + 4.33i)T \)
good5 \( 1 + (-0.165 - 0.936i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.67 + 1.54i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.39 + 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.284 + 0.781i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.14 + 2.63i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (2.45 + 0.432i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (1.95 - 2.32i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.560 + 0.970i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + (1.74 - 4.80i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-2.92 + 0.515i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.23 + 2.66i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (-4.27 - 0.753i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-0.677 + 0.568i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.74 - 9.90i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.32 - 5.30i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.66 - 9.42i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.43 - 0.887i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (3.07 + 1.12i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (10.7 + 6.18i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.254 - 0.698i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.10 + 7.27i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.23654494629986218593947795146, −9.770811255276549608619810762837, −8.724187947796925168198508942880, −7.14197203370614869078441392849, −6.63247249751926096297193754984, −5.74115089740246455765496623873, −4.52749242288288638141596802039, −3.47888808775419049130387568522, −2.78601829598623561915701173683, −0.946676122150415101975791795143, 1.98844367494514572896517367547, 3.44409064853629480960107841659, 4.18539599900585955520947705001, 5.48498543712507291965319552476, 6.13056112629111145301744944278, 6.98502803438341410402508624046, 7.975783084268017101301424460409, 8.993542855636281468717770263144, 9.648314407637496890473752585189, 10.82874391076723548103963134187

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