Properties

Label 2-24-24.11-c1-0-1
Degree $2$
Conductor $24$
Sign $0.816 + 0.577i$
Analytic cond. $0.191640$
Root an. cond. $0.437768$
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  − 1.41i·2-s + (−1 + 1.41i)3-s − 2.00·4-s + (2.00 + 1.41i)6-s + 2.82i·8-s + (−1.00 − 2.82i)9-s − 2.82i·11-s + (2.00 − 2.82i)12-s + 4.00·16-s + 5.65i·17-s + (−4.00 + 1.41i)18-s + 2·19-s − 4.00·22-s + (−4.00 − 2.82i)24-s − 5·25-s + ⋯
L(s)  = 1  − 0.999i·2-s + (−0.577 + 0.816i)3-s − 1.00·4-s + (0.816 + 0.577i)6-s + 1.00i·8-s + (−0.333 − 0.942i)9-s − 0.852i·11-s + (0.577 − 0.816i)12-s + 1.00·16-s + 1.37i·17-s + (−0.942 + 0.333i)18-s + 0.458·19-s − 0.852·22-s + (−0.816 − 0.577i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(0.191640\)
Root analytic conductor: \(0.437768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511280 - 0.162503i\)
\(L(\frac12)\) \(\approx\) \(0.511280 - 0.162503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + (1 - 1.41i)T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−17.71275726124330819883970306751, −16.70501923385416248717129689884, −15.19730047189340621458262065414, −13.75877200386722474683296855826, −12.22728533480882669071683978630, −11.08980142422841485939985229545, −10.05403923244183463898302503844, −8.628873380779201877048120215204, −5.65243967610745928202997489765, −3.79734348170821547926452829026, 5.09127124120506387813062870719, 6.71952475783811132454670618982, 7.86668938059784971565526833646, 9.720994441844088178423615033944, 11.75089254705110530455108571385, 13.07485440210465344807068665664, 14.15078414747634405407289801866, 15.64610061145699713347539417384, 16.80125482013019331873847309096, 17.86271258287136632536477627055

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