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Properties

Label 2-1224-51.50-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.3823805483\)
\(L(\frac12)\) \(\approx\) \(0.3823805483\)
\(L(1)\) 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

−10.39767306844368612725729412196, −9.108580759732845590274149584498, −8.762271147200267393356025636917, −7.80028273994966925569329579545, −7.11853972412115120464203667236, −6.01753732011591840324226853568, −5.09168462835072136516363774183, −4.36632991453827353164468195972, −2.98246259987145635389225049074, −2.24228587948386791577121929546, 0.30799226672292309201947228854, 2.29341822991056717597436260083, 3.55991240279185510432456386374, 4.39834475904926607730509203379, 5.05062291657153624056412159918, 6.55652904619864707357625669315, 7.22604201157980910646828305987, 7.86793445908189252636694017840, 8.571124003580642402683794729419, 9.798387357837920282483225843388

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