Properties

Label 2-2736-3.2-c2-0-70
Degree $2$
Conductor $2736$
Sign $-0.577 - 0.816i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.04i·5-s + 3.07·7-s − 7.06i·11-s − 21.0·13-s − 2.77i·17-s + 4.35·19-s − 18.0i·23-s − 24.5·25-s + 19.2i·29-s − 7.72·31-s − 21.6i·35-s + 7.60·37-s + 36.9i·41-s − 45.9·43-s − 58.6i·47-s + ⋯
L(s)  = 1  − 1.40i·5-s + 0.439·7-s − 0.642i·11-s − 1.61·13-s − 0.163i·17-s + 0.229·19-s − 0.785i·23-s − 0.983·25-s + 0.663i·29-s − 0.249·31-s − 0.619i·35-s + 0.205·37-s + 0.901i·41-s − 1.06·43-s − 1.24i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2215650357\)
\(L(\frac12)\) \(\approx\) \(0.2215650357\)
\(L(2)\) 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 \( 1 \)
19 \( 1 - 4.35T \)
good5 \( 1 + 7.04iT - 25T^{2} \)
7 \( 1 - 3.07T + 49T^{2} \)
11 \( 1 + 7.06iT - 121T^{2} \)
13 \( 1 + 21.0T + 169T^{2} \)
17 \( 1 + 2.77iT - 289T^{2} \)
23 \( 1 + 18.0iT - 529T^{2} \)
29 \( 1 - 19.2iT - 841T^{2} \)
31 \( 1 + 7.72T + 961T^{2} \)
37 \( 1 - 7.60T + 1.36e3T^{2} \)
41 \( 1 - 36.9iT - 1.68e3T^{2} \)
43 \( 1 + 45.9T + 1.84e3T^{2} \)
47 \( 1 + 58.6iT - 2.20e3T^{2} \)
53 \( 1 - 23.4iT - 2.80e3T^{2} \)
59 \( 1 + 31.2iT - 3.48e3T^{2} \)
61 \( 1 + 72.8T + 3.72e3T^{2} \)
67 \( 1 - 86.3T + 4.48e3T^{2} \)
71 \( 1 - 117. iT - 5.04e3T^{2} \)
73 \( 1 + 59.1T + 5.32e3T^{2} \)
79 \( 1 - 42.9T + 6.24e3T^{2} \)
83 \( 1 + 30.4iT - 6.88e3T^{2} \)
89 \( 1 + 41.2iT - 7.92e3T^{2} \)
97 \( 1 + 67.6T + 9.40e3T^{2} \)
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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

−8.299016225669940914491685108047, −7.53940895542700278802594507034, −6.67321724633605245096941911707, −5.57706517038609598320837265592, −4.93959899643279738717676447230, −4.49751335900466768736433382610, −3.25938158426057203288964103326, −2.14384921198571226423960088952, −1.07736409176277999263851783204, −0.05247558985655873924872888076, 1.77772214832314862603937085964, 2.57283619813955173984938424722, 3.39701610781270426389517584005, 4.46978839776210862369533192205, 5.23329926391825172097288688819, 6.20922555421797922119577902092, 7.04076217120199776575580629416, 7.47439106971886623944270868687, 8.133376642215019146175569409841, 9.448556152295028658965574828765

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