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Properties

Label 2-1824-12.11-c1-0-53
Degree $2$
Conductor $1824$
Sign $-0.0640 + 0.997i$
Analytic cond. $14.5647$
Root an. cond. $3.81637$
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  + (−1.30 + 1.14i)3-s − 3.53i·5-s − 3.09i·7-s + (0.383 − 2.97i)9-s + 5.60·11-s + 0.378·13-s + (4.04 + 4.59i)15-s + 1.90i·17-s − i·19-s + (3.54 + 4.02i)21-s + 7.91·23-s − 7.48·25-s + (2.90 + 4.30i)27-s − 9.18i·29-s + 6.82i·31-s + ⋯
L(s)  = 1  + (−0.750 + 0.660i)3-s − 1.58i·5-s − 1.17i·7-s + (0.127 − 0.991i)9-s + 1.69·11-s + 0.105·13-s + (1.04 + 1.18i)15-s + 0.463i·17-s − 0.229i·19-s + (0.772 + 0.878i)21-s + 1.64·23-s − 1.49·25-s + (0.559 + 0.829i)27-s − 1.70i·29-s + 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0640 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0640 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $-0.0640 + 0.997i$
Analytic conductor: \(14.5647\)
Root analytic conductor: \(3.81637\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1824} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :1/2),\ -0.0640 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.434538899\)
\(L(\frac12)\) \(\approx\) \(1.434538899\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 - 1.14i)T \)
19 \( 1 + iT \)
good5 \( 1 + 3.53iT - 5T^{2} \)
7 \( 1 + 3.09iT - 7T^{2} \)
11 \( 1 - 5.60T + 11T^{2} \)
13 \( 1 - 0.378T + 13T^{2} \)
17 \( 1 - 1.90iT - 17T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 + 9.18iT - 29T^{2} \)
31 \( 1 - 6.82iT - 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 - 0.521iT - 41T^{2} \)
43 \( 1 + 6.60iT - 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 8.42iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 - 9.11iT - 67T^{2} \)
71 \( 1 + 1.69T + 71T^{2} \)
73 \( 1 + 2.30T + 73T^{2} \)
79 \( 1 + 5.74iT - 79T^{2} \)
83 \( 1 + 8.83T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.003156918408155296336067485643, −8.670100503643489378496086633947, −7.34081094528930483190411088411, −6.60488170491459366860061872652, −5.70868703893729067228795795301, −4.77817694684175345586167556793, −4.23967913020591036063035118985, −3.59000542794587121863701750427, −1.36329244152818240736818605233, −0.70454512468731196621871621442, 1.39289341443256600109211941087, 2.54075251887986774087021656855, 3.34747296325995142592072688431, 4.67253744568824084875457769910, 5.80743160705311538538673502763, 6.27977128910382794800711332303, 7.02279426748080469957013306974, 7.49958049924618660434565279166, 8.836594447878067938856473075922, 9.330006251677026712427332253952

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