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Properties

Label 2-1824-19.7-c1-0-28
Degree $2$
Conductor $1824$
Sign $-0.0977 + 0.995i$
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  + (−0.5 − 0.866i)3-s + (−1.41 − 2.44i)5-s + 1.82·7-s + (−0.499 + 0.866i)9-s + 4.82·11-s + (1.91 − 3.31i)13-s + (−1.41 + 2.44i)15-s + (2.41 + 4.18i)17-s + (−4 − 1.73i)19-s + (−0.914 − 1.58i)21-s + (3 − 5.19i)23-s + (−1.49 + 2.59i)25-s + 0.999·27-s + (−0.414 + 0.717i)29-s + 5.82·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.632 − 1.09i)5-s + 0.691·7-s + (−0.166 + 0.288i)9-s + 1.45·11-s + (0.530 − 0.919i)13-s + (−0.365 + 0.632i)15-s + (0.585 + 1.01i)17-s + (−0.917 − 0.397i)19-s + (−0.199 − 0.345i)21-s + (0.625 − 1.08i)23-s + (−0.299 + 0.519i)25-s + 0.192·27-s + (−0.0769 + 0.133i)29-s + 1.04·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1824\)    =    \(2^{5} \cdot 3 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(14.5647\)
Root analytic conductor: \(3.81637\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1824} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1824,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.680546863\)
\(L(\frac12)\) \(\approx\) \(1.680546863\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (1.41 + 2.44i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.82T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + (-1.91 + 3.31i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.41 - 4.18i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.414 - 0.717i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.82T + 31T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + (-5.65 - 9.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.82 - 6.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.82 - 10.0i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.82 + 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.08 + 3.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.07 + 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.91 + 5.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.828T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.65 + 14.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.845076120659365044711357469618, −8.084394821973971940056253207313, −7.88705028643887448885910083129, −6.38768619298418151305146160876, −6.10157305394970534832558486900, −4.60253482062301038499148814845, −4.48135792215547657842119113271, −3.10515571488732161494668357063, −1.51496850190413102412707814854, −0.805849509292456709380277384698, 1.27496253422274264087423914952, 2.72124184797562129600787527929, 3.86667794064398268417549298947, 4.22745575085485749481442811833, 5.44051822463823072948121872903, 6.43537865867438145608567024827, 6.98671686632297525853016175456, 7.82193306959276688169879014264, 8.789158085890425785287845405176, 9.472586830882769665161255878243

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