Properties

Label 2-2178-3.2-c2-0-3
Degree $2$
Conductor $2178$
Sign $-0.816 + 0.577i$
Analytic cond. $59.3462$
Root an. cond. $7.70364$
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  − 1.41i·2-s − 2.00·4-s + 9.68i·5-s − 8.70·7-s + 2.82i·8-s + 13.6·10-s − 2.23·13-s + 12.3i·14-s + 4.00·16-s + 6.99i·17-s − 6.91·19-s − 19.3i·20-s + 34.1i·23-s − 68.8·25-s + 3.15i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 1.93i·5-s − 1.24·7-s + 0.353i·8-s + 1.36·10-s − 0.171·13-s + 0.879i·14-s + 0.250·16-s + 0.411i·17-s − 0.363·19-s − 0.968i·20-s + 1.48i·23-s − 2.75·25-s + 0.121i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2178\)    =    \(2 \cdot 3^{2} \cdot 11^{2}\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(59.3462\)
Root analytic conductor: \(7.70364\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2178} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2178,\ (\ :1),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2473210444\)
\(L(\frac12)\) \(\approx\) \(0.2473210444\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 - 9.68iT - 25T^{2} \)
7 \( 1 + 8.70T + 49T^{2} \)
13 \( 1 + 2.23T + 169T^{2} \)
17 \( 1 - 6.99iT - 289T^{2} \)
19 \( 1 + 6.91T + 361T^{2} \)
23 \( 1 - 34.1iT - 529T^{2} \)
29 \( 1 - 48.7iT - 841T^{2} \)
31 \( 1 + 7.91T + 961T^{2} \)
37 \( 1 + 22.1T + 1.36e3T^{2} \)
41 \( 1 + 64.4iT - 1.68e3T^{2} \)
43 \( 1 - 60.7T + 1.84e3T^{2} \)
47 \( 1 - 53.8iT - 2.20e3T^{2} \)
53 \( 1 - 37.7iT - 2.80e3T^{2} \)
59 \( 1 - 46.6iT - 3.48e3T^{2} \)
61 \( 1 + 25.2T + 3.72e3T^{2} \)
67 \( 1 + 15.8T + 4.48e3T^{2} \)
71 \( 1 + 29.9iT - 5.04e3T^{2} \)
73 \( 1 - 32.4T + 5.32e3T^{2} \)
79 \( 1 - 23.4T + 6.24e3T^{2} \)
83 \( 1 + 44.7iT - 6.88e3T^{2} \)
89 \( 1 + 118. iT - 7.92e3T^{2} \)
97 \( 1 + 157.T + 9.40e3T^{2} \)
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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

−9.551955958522009002308391944852, −8.865518684671621666112314004003, −7.49264639083782818183782502439, −7.13135466045663505430725674123, −6.21825918185623323619609878470, −5.60179434391594907256023802710, −4.06034703196046891392435224133, −3.32127545524278867183323799985, −2.87267307863490094008036961946, −1.79220137213635586703884392415, 0.079372996611393857147562001010, 0.820740660022706072552508742637, 2.35779806487288083796362643482, 3.79372372635846106985503168698, 4.51276934050306117474378160086, 5.23506949902301015754437741807, 6.08601336399113725452268812854, 6.70155784394622444819399158257, 7.86691267183561394932473568972, 8.372174603662678246966385476563

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