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Properties

Label 2-1224-1224.371-c0-0-0
Degree $2$
Conductor $1224$
Sign $0.669 + 0.743i$
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  + (−0.991 − 0.130i)2-s + (−0.965 + 0.258i)3-s + (0.965 + 0.258i)4-s + (0.991 − 0.130i)6-s + (−0.923 − 0.382i)8-s + (0.866 − 0.499i)9-s + (−0.608 − 0.206i)11-s − 12-s + (0.866 + 0.5i)16-s + (0.130 − 0.991i)17-s + (−0.923 + 0.382i)18-s + (0.478 − 0.198i)19-s + (0.576 + 0.284i)22-s + (0.991 + 0.130i)24-s + (−0.608 − 0.793i)25-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)2-s + (−0.965 + 0.258i)3-s + (0.965 + 0.258i)4-s + (0.991 − 0.130i)6-s + (−0.923 − 0.382i)8-s + (0.866 − 0.499i)9-s + (−0.608 − 0.206i)11-s − 12-s + (0.866 + 0.5i)16-s + (0.130 − 0.991i)17-s + (−0.923 + 0.382i)18-s + (0.478 − 0.198i)19-s + (0.576 + 0.284i)22-s + (0.991 + 0.130i)24-s + (−0.608 − 0.793i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.669 + 0.743i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.669 + 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4509417576\)
\(L(\frac12)\) \(\approx\) \(0.4509417576\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.991 + 0.130i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
17 \( 1 + (-0.130 + 0.991i)T \)
good5 \( 1 + (0.608 + 0.793i)T^{2} \)
7 \( 1 + (-0.608 + 0.793i)T^{2} \)
11 \( 1 + (0.608 + 0.206i)T + (0.793 + 0.608i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.478 + 0.198i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.130 - 0.991i)T^{2} \)
29 \( 1 + (0.991 - 0.130i)T^{2} \)
31 \( 1 + (0.793 - 0.608i)T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (-1.88 - 0.123i)T + (0.991 + 0.130i)T^{2} \)
43 \( 1 + (0.207 - 0.158i)T + (0.258 - 0.965i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.158 + 1.20i)T + (-0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.608 - 0.793i)T^{2} \)
67 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (-1.25 + 0.835i)T + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.793 + 0.608i)T^{2} \)
83 \( 1 + (-0.241 + 1.83i)T + (-0.965 - 0.258i)T^{2} \)
89 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
97 \( 1 + (1.78 - 0.117i)T + (0.991 - 0.130i)T^{2} \)
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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

−9.753015486394073039400639355269, −9.322233620845137964358007779971, −8.133275988050619969432885887141, −7.42772393303459292466267452461, −6.57789763486594559736276964937, −5.75653469765220149614818946087, −4.84843009184817568511887885905, −3.54929826842654646674021804065, −2.30908045068284068348947637698, −0.69581027886490647354479091577, 1.23145743650250129304671301170, 2.43415030721846935028674578172, 3.98357207890666693463558005762, 5.38485651768080113901011527534, 5.88903919701210696464437038219, 6.87294599376140987940491254906, 7.59448211571130127789693230808, 8.234710863882040696483344776866, 9.394106739291021550022309737894, 10.03077344114346344544351329392

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