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Properties

Label 2-552-552.413-c1-0-34
Degree $2$
Conductor $552$
Sign $0.995 + 0.0961i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 0.921i)2-s + (−1.37 − 1.05i)3-s + (0.300 − 1.97i)4-s + (2.44 − 0.133i)6-s + (1.5 + 2.39i)8-s + (0.771 + 2.89i)9-s + (−2.49 + 2.39i)12-s + 1.57i·13-s + (−3.81 − 1.18i)16-s + (−3.49 − 2.39i)18-s − 4.79i·23-s + (0.471 − 4.87i)24-s + 5·25-s + (−1.45 − 1.69i)26-s + (1.99 − 4.79i)27-s + ⋯
L(s)  = 1  + (−0.758 + 0.651i)2-s + (−0.792 − 0.609i)3-s + (0.150 − 0.988i)4-s + (0.998 − 0.0546i)6-s + (0.530 + 0.847i)8-s + (0.257 + 0.966i)9-s + (−0.721 + 0.692i)12-s + 0.437i·13-s + (−0.954 − 0.297i)16-s + (−0.824 − 0.565i)18-s − 0.999i·23-s + (0.0961 − 0.995i)24-s + 25-s + (−0.284 − 0.331i)26-s + (0.384 − 0.922i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.995 + 0.0961i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.995 + 0.0961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.738518 - 0.0355904i\)
\(L(\frac12)\) \(\approx\) \(0.738518 - 0.0355904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 - 0.921i)T \)
3 \( 1 + (1.37 + 1.05i)T \)
23 \( 1 + 4.79iT \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 1.57iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 - 5.83T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 2.64iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 13.7iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 1.04iT - 71T^{2} \)
73 \( 1 - 9.44T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.61500376899758333017127129940, −10.01995565830155194288556899248, −8.771377722036705156063181887663, −8.097619237290497003096952766326, −6.94667632805142237860823011588, −6.54166394833949152589158578754, −5.44629330746149902717102522335, −4.54878464037263581037634121409, −2.32357394428477704703195923118, −0.841341606211256290021729053829, 0.995759762935498704351035542620, 2.85693456509797238924217551768, 3.99270340265234042573919208525, 5.06576330830332205767191923101, 6.29970039389590919865636268675, 7.28464936889020020673891193475, 8.376567999345587800449744105865, 9.288555547390402383817460466191, 10.04962926665554413934499280760, 10.72854157813298672044366949520

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