Properties

Label 2-4560-76.75-c1-0-36
Degree $2$
Conductor $4560$
Sign $0.989 - 0.141i$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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-s − 5-s + 1.63i·7-s + 9-s + 1.02i·11-s + 1.63i·13-s + 15-s + 3.31·17-s + (4.31 − 0.616i)19-s − 1.63i·21-s − 8.09i·23-s + 25-s − 27-s + 1.02i·29-s − 6.30·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.619i·7-s + 0.333·9-s + 0.308i·11-s + 0.454i·13-s + 0.258·15-s + 0.804·17-s + (0.989 − 0.141i)19-s − 0.357i·21-s − 1.68i·23-s + 0.200·25-s − 0.192·27-s + 0.189i·29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.345395145\)
\(L(\frac12)\) \(\approx\) \(1.345395145\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
19 \( 1 + (-4.31 + 0.616i)T \)
good7 \( 1 - 1.63iT - 7T^{2} \)
11 \( 1 - 1.02iT - 11T^{2} \)
13 \( 1 - 1.63iT - 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
23 \( 1 + 8.09iT - 23T^{2} \)
29 \( 1 - 1.02iT - 29T^{2} \)
31 \( 1 + 6.30T + 31T^{2} \)
37 \( 1 + 9.11iT - 37T^{2} \)
41 \( 1 - 2.25iT - 41T^{2} \)
43 \( 1 + 9.11iT - 43T^{2} \)
47 \( 1 - 6.85iT - 47T^{2} \)
53 \( 1 - 8.09iT - 53T^{2} \)
59 \( 1 + 6.63T + 59T^{2} \)
61 \( 1 + 4.30T + 61T^{2} \)
67 \( 1 - 0.989T + 67T^{2} \)
71 \( 1 + 6.63T + 71T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 + 1.61T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + 5.22iT - 89T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−8.327111663331528540895861201684, −7.46661862982695984432640617279, −7.00752767994055434456627787293, −6.03672566031754810033255819734, −5.46608465487536351061739965861, −4.67175610996637330056584487220, −3.90530412683996598864837199260, −2.92230921525461363192249061481, −1.91941063877319390263066336520, −0.66013527500673227512376064884, 0.70632541317693189676312896758, 1.62269645581423776388055752132, 3.25149379641849326733817922861, 3.58314662754336707994178878673, 4.69774179592031188033722466169, 5.39519313063153846780594766293, 6.00999253524596650777242094759, 6.99932558034886080422295636961, 7.58928300679870304181948758974, 8.039541230790569736320935222199

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