Properties

Label 2-4608-8.5-c1-0-9
Degree $2$
Conductor $4608$
Sign $-i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  − 3.41i·5-s + 1.41·7-s + 4.82i·11-s + 0.828i·13-s − 4.82·17-s − 2.82i·19-s + 1.17·23-s − 6.65·25-s + 7.41i·29-s + 7.07·31-s − 4.82i·35-s + 11.6i·37-s − 10.4·41-s + 6.82i·43-s − 12.4·47-s + ⋯
L(s)  = 1  − 1.52i·5-s + 0.534·7-s + 1.45i·11-s + 0.229i·13-s − 1.17·17-s − 0.648i·19-s + 0.244·23-s − 1.33·25-s + 1.37i·29-s + 1.27·31-s − 0.816i·35-s + 1.91i·37-s − 1.63·41-s + 1.04i·43-s − 1.82·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9757817210\)
\(L(\frac12)\) \(\approx\) \(0.9757817210\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 3.41iT - 5T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 - 0.828iT - 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
29 \( 1 - 7.41iT - 29T^{2} \)
31 \( 1 - 7.07T + 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 6.82iT - 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + 1.75iT - 53T^{2} \)
59 \( 1 + 1.65iT - 59T^{2} \)
61 \( 1 + 0.343iT - 61T^{2} \)
67 \( 1 - 5.65iT - 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 17.4T + 79T^{2} \)
83 \( 1 + 8.82iT - 83T^{2} \)
89 \( 1 - 5.31T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−8.465084155785790375799710635503, −8.034037510315620950080834453992, −6.93749234352531580291510902669, −6.51514223805842128459636043592, −5.16463715366590806652447874952, −4.66505766908542553625372232167, −4.53448733097734743451280091338, −3.09266304564396446584437147122, −1.86621062450103178696686087945, −1.28678106560078068432255222913, 0.26067975005371122172957653928, 1.85686991146537560999852054344, 2.74935782179760142071000481167, 3.43643937129051155938965614082, 4.23561029898232362498921459998, 5.32712457680489653151795935294, 6.14582454852208633720049649897, 6.57566729622333213746312434742, 7.39148727268840743730441597010, 8.179955948294707955701986541478

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