Properties

Label 2-1224-51.50-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.816 - 0.577i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 1.41i·7-s + 11-s − 13-s + 17-s − 19-s − 23-s + 1.41i·31-s + 1.41i·35-s − 1.41i·37-s + 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯
L(s)  = 1  + 5-s + 1.41i·7-s + 11-s − 13-s + 17-s − 19-s − 23-s + 1.41i·31-s + 1.41i·35-s − 1.41i·37-s + 41-s + 43-s − 1.41i·47-s − 1.00·49-s − 1.41i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248053139\)
\(L(\frac12)\) \(\approx\) \(1.248053139\)
\(L(1)\) 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 \( 1 \)
17 \( 1 - T \)
good5 \( 1 - T + T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.867407862659683404900532480026, −9.194040615529656349970865681908, −8.641408709000047708409357887267, −7.54386169340081588467820835961, −6.46782998805953865523129560527, −5.80471549177089831131357871858, −5.16388001754828158897653412547, −3.89187089406014943455718454888, −2.54406568015478086719457527751, −1.83283734780885241631386477879, 1.25591108254819659868895857152, 2.47849886880256181301006945127, 3.89756949744959439524906263284, 4.53235417807861078972735299753, 5.83447815758222056735076990041, 6.43600279256790155076679622790, 7.41320994977026549169208484515, 8.044330708331787325220962429865, 9.394340823552955517760951048873, 9.780824532793101687100032429123

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