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Properties

Label 2-1104-12.11-c1-0-18
Degree $2$
Conductor $1104$
Sign $0.776 - 0.630i$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
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  + (1.61 + 0.618i)3-s − 1.54i·5-s + 2.42i·7-s + (2.23 + 2.00i)9-s − 2.73·11-s + 5.46·13-s + (0.956 − 2.50i)15-s + 4.19i·17-s − 1.54i·19-s + (−1.49 + 3.91i)21-s + 23-s + 2.60·25-s + (2.38 + 4.61i)27-s − 4.47i·29-s + 5.60i·31-s + ⋯
L(s)  = 1  + (0.934 + 0.356i)3-s − 0.692i·5-s + 0.914i·7-s + (0.745 + 0.666i)9-s − 0.823·11-s + 1.51·13-s + (0.246 − 0.646i)15-s + 1.01i·17-s − 0.355i·19-s + (−0.326 + 0.854i)21-s + 0.208·23-s + 0.521·25-s + (0.458 + 0.888i)27-s − 0.830i·29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.776 - 0.630i$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ 0.776 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.327433081\)
\(L(\frac12)\) \(\approx\) \(2.327433081\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.61 - 0.618i)T \)
23 \( 1 - T \)
good5 \( 1 + 1.54iT - 5T^{2} \)
7 \( 1 - 2.42iT - 7T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 + 1.54iT - 19T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 5.60iT - 31T^{2} \)
37 \( 1 + 0.872T + 37T^{2} \)
41 \( 1 - 4.22iT - 41T^{2} \)
43 \( 1 - 4.82iT - 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 0.539iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 10.8iT - 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−9.813454023384783421541489746644, −8.771488221222345684745837824674, −8.592274548635011660149557214870, −7.84004489846406748858491125347, −6.53762136128895061291168106231, −5.54987292427662674889016388211, −4.70586749433619754955395987297, −3.66312989653876565325226297933, −2.67997666703757770136865622022, −1.49318315543339516613449170341, 1.06240381965662261696535337922, 2.51669184782693725006538785910, 3.40260058635909592038542831656, 4.20233075123798277672306504853, 5.58605308331502016835471056230, 6.75116344406265250173656163520, 7.25147023980334536304586152858, 8.059620553417496076115422190250, 8.845717143652845255002637499788, 9.741614505194210314999402776702

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