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Properties

Label 2-1488-31.5-c1-0-19
Degree $2$
Conductor $1488$
Sign $-0.695 + 0.718i$
Analytic cond. $11.8817$
Root an. cond. $3.44698$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−1 − 1.73i)5-s + (−2.5 + 4.33i)7-s + (−0.499 − 0.866i)9-s + (1.5 + 2.59i)11-s + 1.99·15-s + (−3 + 5.19i)17-s + (3 − 5.19i)19-s + (−2.5 − 4.33i)21-s + (0.500 − 0.866i)25-s + 0.999·27-s − 3·29-s + (−3.5 − 4.33i)31-s − 3·33-s + 10·35-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.944 + 1.63i)7-s + (−0.166 − 0.288i)9-s + (0.452 + 0.783i)11-s + 0.516·15-s + (−0.727 + 1.26i)17-s + (0.688 − 1.19i)19-s + (−0.545 − 0.944i)21-s + (0.100 − 0.173i)25-s + 0.192·27-s − 0.557·29-s + (−0.628 − 0.777i)31-s − 0.522·33-s + 1.69·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1488\)    =    \(2^{4} \cdot 3 \cdot 31\)
Sign: $-0.695 + 0.718i$
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: \(1\)
Selberg data: \((2,\ 1488,\ (\ :1/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 4.33i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8 + 13.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.5 - 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 19T + 97T^{2} \)
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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

−9.153673149410979115889597609400, −8.767341415528167341502239949644, −7.67244069435879189612221353971, −6.50873312823639780429910019951, −5.90153276319626711006001732227, −4.97068752152776762962527411895, −4.21643474825327726722132029789, −3.12165467101626265349414709420, −1.95783712826099230772220978265, 0, 1.28243852987123691967920215766, 3.12949915314682400216547233753, 3.54162237954479477473247677304, 4.69154069124119046346019414988, 5.97238991762956117799807555645, 6.76572691196435997534573255817, 7.20319955575012453977850619440, 7.87520613485329641960022481136, 9.047737945493122687598691912892

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