Properties

Label 2-72-24.11-c1-0-2
Degree $2$
Conductor $72$
Sign $0.816 - 0.577i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
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  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s − 2.44·5-s − 3.46i·7-s + 2.82i·8-s + (−2.99 − 1.73i)10-s − 2.82i·11-s + 3.46i·13-s + (2.44 − 4.24i)14-s + (−2.00 + 3.46i)16-s + 1.41i·17-s − 4·19-s + (−2.44 − 4.24i)20-s + (2.00 − 3.46i)22-s + 4.89·23-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.499 + 0.866i)4-s − 1.09·5-s − 1.30i·7-s + 0.999i·8-s + (−0.948 − 0.547i)10-s − 0.852i·11-s + 0.960i·13-s + (0.654 − 1.13i)14-s + (−0.500 + 0.866i)16-s + 0.342i·17-s − 0.917·19-s + (−0.547 − 0.948i)20-s + (0.426 − 0.738i)22-s + 1.02·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17864 + 0.374617i\)
\(L(\frac12)\) \(\approx\) \(1.17864 + 0.374617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
good5 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−14.64323898232442380148507530619, −13.79861840819444352479777612259, −12.76446738386912383989846350350, −11.52594583535222412157635773597, −10.76323621748918301563556176632, −8.607130072939384169190507963813, −7.50609701678124924502368753315, −6.50584799520617170514405332178, −4.55316731769321763434506621386, −3.58544198419714711481216880062, 2.72983818913073669911553254582, 4.40620656724501533896655294351, 5.74686033431319537469101676940, 7.35520276212401321045431744997, 8.883495842307213316956895826495, 10.37566478281870268857717604396, 11.60567828753957682498601009338, 12.29443882612856770308922875106, 13.14630267806537297381597806815, 15.00296167235404046287622062346

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