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Properties

Label 2-4680-5.4-c1-0-55
Degree $2$
Conductor $4680$
Sign $0.590 + 0.807i$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
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  + (−1.32 − 1.80i)5-s + 4.64·11-s − i·13-s − 4.24i·17-s + 6.24·19-s + 2.24i·23-s + (−1.51 + 4.76i)25-s − 9.21·29-s + 9.28·31-s + 7.28i·37-s + 5.67·41-s + 4.24i·43-s − 2.88i·47-s + 7·49-s − 9.21i·53-s + ⋯
L(s)  = 1  + (−0.590 − 0.807i)5-s + 1.39·11-s − 0.277i·13-s − 1.03i·17-s + 1.43·19-s + 0.469i·23-s + (−0.303 + 0.952i)25-s − 1.71·29-s + 1.66·31-s + 1.19i·37-s + 0.885·41-s + 0.648i·43-s − 0.421i·47-s + 49-s − 1.26i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.590 + 0.807i$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4680} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ 0.590 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.935877096\)
\(L(\frac12)\) \(\approx\) \(1.935877096\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1.32 + 1.80i)T \)
13 \( 1 + iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 4.64T + 11T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 - 2.24iT - 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 - 9.28T + 31T^{2} \)
37 \( 1 - 7.28iT - 37T^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + 2.88iT - 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 - 0.969T + 61T^{2} \)
67 \( 1 + 1.93iT - 67T^{2} \)
71 \( 1 + 5.60T + 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 3.67iT - 83T^{2} \)
89 \( 1 + 9.67T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.230697021696547120823972166986, −7.43208548335375710538221930372, −6.95005375838309732910075642872, −5.89183448816922971218214625537, −5.22374879187697506366668300811, −4.43745917766107825015043634404, −3.71361675055774189050791387533, −2.91116198848940890657210184104, −1.48852049399851799998283749392, −0.71982684269182858581747500371, 0.948629858708896455709054543124, 2.10553046701344435642962485854, 3.16240868630571458552652351563, 3.91448958986992446456638781821, 4.39989038625850932156593794582, 5.74500155277341439032436228531, 6.20086024720175438117583436428, 7.14800034352124969813701017431, 7.46222732410433249347188830150, 8.411724813109780973455536093848

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