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Properties

Label 2-1512-8.5-c1-0-15
Degree $2$
Conductor $1512$
Sign $-0.587 - 0.809i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−0.437 + 1.34i)2-s + (−1.61 − 1.17i)4-s + 0.0549i·5-s − 7-s + (2.28 − 1.66i)8-s + (−0.0738 − 0.0240i)10-s − 2.63i·11-s + 3.67i·13-s + (0.437 − 1.34i)14-s + (1.23 + 3.80i)16-s − 3.16·17-s − 3.07i·19-s + (0.0645 − 0.0888i)20-s + (3.54 + 1.15i)22-s − 2.86·23-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + 0.0245i·5-s − 0.377·7-s + (0.809 − 0.587i)8-s + (−0.0233 − 0.00759i)10-s − 0.794i·11-s + 1.02i·13-s + (0.116 − 0.359i)14-s + (0.309 + 0.951i)16-s − 0.766·17-s − 0.706i·19-s + (0.0144 − 0.0198i)20-s + (0.755 + 0.245i)22-s − 0.598·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.587 - 0.809i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.587 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9622712626\)
\(L(\frac12)\) \(\approx\) \(0.9622712626\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.437 - 1.34i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 0.0549iT - 5T^{2} \)
11 \( 1 + 2.63iT - 11T^{2} \)
13 \( 1 - 3.67iT - 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 3.07iT - 19T^{2} \)
23 \( 1 + 2.86T + 23T^{2} \)
29 \( 1 - 10.1iT - 29T^{2} \)
31 \( 1 - 9.32T + 31T^{2} \)
37 \( 1 + 0.774iT - 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 - 9.98iT - 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 + 3.39iT - 53T^{2} \)
59 \( 1 - 6.93iT - 59T^{2} \)
61 \( 1 - 8.35iT - 61T^{2} \)
67 \( 1 - 8.93iT - 67T^{2} \)
71 \( 1 + 4.28T + 71T^{2} \)
73 \( 1 - 8.38T + 73T^{2} \)
79 \( 1 + 3.03T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.471498827160790507664057851020, −8.796993605027250064616428443910, −8.342738709840562675260596052991, −7.05517806966758805405351833951, −6.72223940089848188851340853836, −5.84867384520662876991251251283, −4.85867835928508413788939144738, −4.08066832386461897621928812083, −2.78307014194898496347404704229, −1.12854742463712650560626280662, 0.49620239928811701615723926511, 2.00732282010450982891365552208, 2.89916159700506457097558231149, 3.97047013862448025112793185121, 4.74133891095232833868209962658, 5.81741984976955957917290179568, 6.88466412955662931189951270763, 7.889456566797553049411127464396, 8.444625445476587149747572191407, 9.406669353958596429892476814765

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