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Properties

Label 2-4560-76.75-c1-0-44
Degree $2$
Conductor $4560$
Sign $0.993 - 0.116i$
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 − 0.238i·7-s + 9-s − 5.02i·11-s + 5.35i·13-s + 15-s + 4.36·17-s + (−1.72 + 4.00i)19-s − 0.238i·21-s − 5.83i·23-s + 25-s + 27-s + 8.48i·29-s + 5.65·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.0900i·7-s + 0.333·9-s − 1.51i·11-s + 1.48i·13-s + 0.258·15-s + 1.05·17-s + (−0.395 + 0.918i)19-s − 0.0519i·21-s − 1.21i·23-s + 0.200·25-s + 0.192·27-s + 1.57i·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.993 - 0.116i$
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: \((2,\ 4560,\ (\ :1/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.811712279\)
\(L(\frac12)\) \(\approx\) \(2.811712279\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 4.00i)T \)
good7 \( 1 + 0.238iT - 7T^{2} \)
11 \( 1 + 5.02iT - 11T^{2} \)
13 \( 1 - 5.35iT - 13T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
23 \( 1 + 5.83iT - 23T^{2} \)
29 \( 1 - 8.48iT - 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 6.51iT - 37T^{2} \)
41 \( 1 + 5.59iT - 41T^{2} \)
43 \( 1 - 6.99iT - 43T^{2} \)
47 \( 1 - 0.328iT - 47T^{2} \)
53 \( 1 + 9.29iT - 53T^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 10.6T + 67T^{2} \)
71 \( 1 - 3.29T + 71T^{2} \)
73 \( 1 - 8.74T + 73T^{2} \)
79 \( 1 - 7.52T + 79T^{2} \)
83 \( 1 + 7.08iT - 83T^{2} \)
89 \( 1 - 5.67iT - 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−8.448399246467151983804662297693, −7.80318107283899072680387011066, −6.69652033891841407254532259794, −6.34889747159445668904214066278, −5.42638318726411018762776815867, −4.55355860326988884180977299486, −3.64726892259997047549107507226, −2.99630781858422272482279876582, −1.95306712868375394459204726142, −1.00127510323370610131652399167, 0.888499468461333493772975611178, 2.10211499043116180198414601239, 2.74193076256271822741663916524, 3.69692954471574781544639295666, 4.60197245997931610688968839831, 5.36643963064693939983698017563, 6.06154785097775432232439313340, 7.05645571698631010566093456040, 7.68990688101864702754605991567, 8.123302726876301900097307288099

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