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Properties

Label 2-1224-408.389-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.637 - 0.770i$
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 + (1.70 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−1.70 + 0.707i)14-s − 1.00·16-s − i·17-s + (0.707 + 0.292i)23-s + (−0.707 + 0.707i)25-s + (0.707 − 1.70i)28-s + (−0.707 − 1.70i)31-s + (0.707 − 0.707i)32-s + (0.707 + 0.707i)34-s + (−0.707 + 1.70i)41-s + (−0.707 + 0.292i)46-s + 1.41·47-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (1.70 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−1.70 + 0.707i)14-s − 1.00·16-s − i·17-s + (0.707 + 0.292i)23-s + (−0.707 + 0.707i)25-s + (0.707 − 1.70i)28-s + (−0.707 − 1.70i)31-s + (0.707 − 0.707i)32-s + (0.707 + 0.707i)34-s + (−0.707 + 1.70i)41-s + (−0.707 + 0.292i)46-s + 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8964122017\)
\(L(\frac12)\) \(\approx\) \(0.8964122017\)
\(L(1)\) 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 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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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

−9.678356416078570515628099127452, −9.190682963115202301233238665847, −8.260944905689909960277755999715, −7.75227467163108599076983390072, −6.97303473770196061275512829990, −5.72076260195449533797532406494, −5.23450734175194440850925135266, −4.32999604909393548255571015912, −2.47962352797793804133462338870, −1.41688532452101076396950036734, 1.27643285912743640047806963889, 2.18712915336866270881471041127, 3.66391248458661202563698091140, 4.42023273534952290825851311649, 5.39520611155340013230683026039, 6.89058256931300398219685241970, 7.54519920941229657910460169511, 8.419184524582442586021521033471, 8.771639571424078202412513634073, 10.03030208758591753728528364140

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