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Properties

Label 2-1488-31.5-c1-0-16
Degree $2$
Conductor $1488$
Sign $0.654 + 0.755i$
Analytic cond. $11.8817$
Root an. cond. $3.44698$
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  + (−0.5 + 0.866i)3-s + (−1.41 − 2.44i)5-s + (−0.207 + 0.358i)7-s + (−0.499 − 0.866i)9-s + (−0.207 − 0.358i)11-s + (2.82 + 4.89i)13-s + 2.82·15-s + (−1.12 + 1.94i)17-s + (1.29 − 2.23i)19-s + (−0.207 − 0.358i)21-s − 2.24·23-s + (−1.49 + 2.59i)25-s + 0.999·27-s + 0.171·29-s + (4.62 − 3.10i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.632 − 1.09i)5-s + (−0.0782 + 0.135i)7-s + (−0.166 − 0.288i)9-s + (−0.0624 − 0.108i)11-s + (0.784 + 1.35i)13-s + 0.730·15-s + (−0.271 + 0.471i)17-s + (0.296 − 0.513i)19-s + (−0.0451 − 0.0782i)21-s − 0.467·23-s + (−0.299 + 0.519i)25-s + 0.192·27-s + 0.0318·29-s + (0.830 − 0.557i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1488\)    =    \(2^{4} \cdot 3 \cdot 31\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(11.8817\)
Root analytic conductor: \(3.44698\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1488} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1488,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160916248\)
\(L(\frac12)\) \(\approx\) \(1.160916248\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-4.62 + 3.10i)T \)
good5 \( 1 + (1.41 + 2.44i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.207 - 0.358i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.207 + 0.358i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.82 - 4.89i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.12 - 1.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.29 + 2.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.24T + 23T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
37 \( 1 + (-3.53 + 6.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.70 + 8.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.82 + 8.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.242T + 47T^{2} \)
53 \( 1 + (4.74 + 8.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.07T + 61T^{2} \)
67 \( 1 + (1.12 + 1.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.82 - 8.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.24 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.24 - 3.88i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.20 + 5.55i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−9.146530845899590752737021538507, −8.786457574925463805811800235958, −7.980694828077344972621383823070, −6.88797675696361748986231281600, −6.01451933439019334914962700124, −5.09842083832086547176325202635, −4.24474240989474279061448811486, −3.72820295444329544171281331480, −2.06989161251263185406632715746, −0.59289027374244386412737451672, 1.06376812061292051967001326328, 2.72849069355326144985855236040, 3.35564094436312134144350555382, 4.52606458066146320925009693999, 5.70727923063905634947460346717, 6.40956556482376266024534599193, 7.20542239069680623313822639096, 7.902497670854188430647443246539, 8.494982980874143858076846105085, 9.883139633949237029333438668838

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