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Properties

Label 2-1224-408.53-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.992 - 0.123i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.292 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.292 + 0.707i)14-s − 1.00·16-s + i·17-s + (0.707 − 1.70i)23-s + (0.707 − 0.707i)25-s + (−0.707 − 0.292i)28-s + (0.707 − 0.292i)31-s + (0.707 − 0.707i)32-s + (−0.707 − 0.707i)34-s + (−0.707 − 0.292i)41-s + (0.707 + 1.70i)46-s + 1.41·47-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.292 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.292 + 0.707i)14-s − 1.00·16-s + i·17-s + (0.707 − 1.70i)23-s + (0.707 − 0.707i)25-s + (−0.707 − 0.292i)28-s + (0.707 − 0.292i)31-s + (0.707 − 0.707i)32-s + (−0.707 − 0.707i)34-s + (−0.707 − 0.292i)41-s + (0.707 + 1.70i)46-s + 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.992 - 0.123i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.992 - 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7772822882\)
\(L(\frac12)\) \(\approx\) \(0.7772822882\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
17 \( 1 - iT \)
good5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
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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

−10.03425303112569880684279239439, −8.822013845038403151276421666823, −8.418662595896370726062927564291, −7.49017128952355948499814889321, −6.73099442433783533253393683257, −6.02113414873382134022124031202, −4.88091610085279321244959937580, −4.12856856486925337428574305214, −2.47728242907352964835070920247, −1.03229254108491104713767584095, 1.35853910772450794100222788865, 2.60545879156263384753211016962, 3.43768132781447198117282409879, 4.72136875344081504756053357285, 5.58267680974852155527728620356, 6.94398839062405278344096648261, 7.53993954509601062928542926216, 8.556650676027506687829671233777, 9.128629864438228053108081657670, 9.802280959388450226781267171442

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