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Properties

Label 2-684-12.11-c1-0-14
Degree $2$
Conductor $684$
Sign $0.944 + 0.329i$
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  + (0.647 − 1.25i)2-s + (−1.16 − 1.62i)4-s + 1.07i·5-s + 3.32i·7-s + (−2.79 + 0.407i)8-s + (1.35 + 0.697i)10-s + 1.96·11-s + 5.47·13-s + (4.17 + 2.15i)14-s + (−1.29 + 3.78i)16-s + 3.80i·17-s − i·19-s + (1.75 − 1.25i)20-s + (1.26 − 2.46i)22-s + 0.563·23-s + ⋯
L(s)  = 1  + (0.457 − 0.889i)2-s + (−0.581 − 0.813i)4-s + 0.482i·5-s + 1.25i·7-s + (−0.989 + 0.144i)8-s + (0.428 + 0.220i)10-s + 0.591·11-s + 1.51·13-s + (1.11 + 0.575i)14-s + (−0.324 + 0.945i)16-s + 0.922i·17-s − 0.229i·19-s + (0.392 − 0.280i)20-s + (0.270 − 0.525i)22-s + 0.117·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.944 + 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82830 - 0.309441i\)
\(L(\frac12)\) \(\approx\) \(1.82830 - 0.309441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.647 + 1.25i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 - 1.07iT - 5T^{2} \)
7 \( 1 - 3.32iT - 7T^{2} \)
11 \( 1 - 1.96T + 11T^{2} \)
13 \( 1 - 5.47T + 13T^{2} \)
17 \( 1 - 3.80iT - 17T^{2} \)
23 \( 1 - 0.563T + 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 - 7.19iT - 31T^{2} \)
37 \( 1 - 4.71T + 37T^{2} \)
41 \( 1 + 2.44iT - 41T^{2} \)
43 \( 1 - 11.9iT - 43T^{2} \)
47 \( 1 + 4.38T + 47T^{2} \)
53 \( 1 + 3.92iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 6.74T + 61T^{2} \)
67 \( 1 + 3.40iT - 67T^{2} \)
71 \( 1 - 0.390T + 71T^{2} \)
73 \( 1 - 6.10T + 73T^{2} \)
79 \( 1 + 8.10iT - 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + 9.56T + 97T^{2} \)
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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

−10.73076442706393653472447433837, −9.624670503190844355716525640494, −8.893520566020089490712983245552, −8.162910050566294514046802949573, −6.29336649783997067317380439670, −6.11807104006195312975598675834, −4.79429583433595873893658984629, −3.67076307116592458701350810936, −2.74280316398384270162135390761, −1.49278851099314479676925060309, 1.01210697897190752759051841285, 3.34080780317636548407264296138, 4.13468176951841739632301290352, 5.03891058662985970142932516071, 6.16383886892702248768517324007, 6.96044646019134461657538805982, 7.74007716993396418191364360281, 8.735638266150534177034984136792, 9.331608408506144494226649369585, 10.58778658487997026721219869997

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