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Properties

Label 2-1008-7.3-c2-0-25
Degree $2$
Conductor $1008$
Sign $-0.232 + 0.972i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.18 + 1.83i)5-s + (−2.47 + 6.54i)7-s + (2.59 − 4.49i)11-s + 2.95i·13-s + (−17.1 − 9.88i)17-s + (−0.805 + 0.465i)19-s + (−1.58 − 2.74i)23-s + (−5.75 + 9.96i)25-s − 31.5·29-s + (35.9 + 20.7i)31-s + (−4.16 − 25.3i)35-s + (−5.95 − 10.3i)37-s − 60.5i·41-s + 68.8·43-s + (20.8 − 12.0i)47-s + ⋯
L(s)  = 1  + (−0.636 + 0.367i)5-s + (−0.353 + 0.935i)7-s + (0.235 − 0.408i)11-s + 0.227i·13-s + (−1.00 − 0.581i)17-s + (−0.0423 + 0.0244i)19-s + (−0.0688 − 0.119i)23-s + (−0.230 + 0.398i)25-s − 1.08·29-s + (1.15 + 0.669i)31-s + (−0.119 − 0.725i)35-s + (−0.160 − 0.278i)37-s − 1.47i·41-s + 1.60·43-s + (0.443 − 0.256i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.232 + 0.972i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ -0.232 + 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5417700283\)
\(L(\frac12)\) \(\approx\) \(0.5417700283\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (2.47 - 6.54i)T \)
good5 \( 1 + (3.18 - 1.83i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.59 + 4.49i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 2.95iT - 169T^{2} \)
17 \( 1 + (17.1 + 9.88i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (0.805 - 0.465i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (1.58 + 2.74i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 31.5T + 841T^{2} \)
31 \( 1 + (-35.9 - 20.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (5.95 + 10.3i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 60.5iT - 1.68e3T^{2} \)
43 \( 1 - 68.8T + 1.84e3T^{2} \)
47 \( 1 + (-20.8 + 12.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-33.1 + 57.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (79.4 + 45.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (10.2 - 5.90i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-46.3 + 80.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 61.0T + 5.04e3T^{2} \)
73 \( 1 + (119. + 68.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (63.7 + 110. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 24.2iT - 6.88e3T^{2} \)
89 \( 1 + (24.3 - 14.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 49.2iT - 9.40e3T^{2} \)
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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

−9.247161267686350829845349929482, −8.912372129512304345961887250910, −7.84414923996722333622150072060, −6.98213414091970434229100252340, −6.17055601779249333687336096155, −5.23155963438742703871308982415, −4.08743941562133788072718200369, −3.13400671646181443941646026390, −2.07794837142326738436265343754, −0.18617342691626846002820147242, 1.14600943787852104378111501870, 2.69062573140262256551890066385, 4.16810277684973071427965178122, 4.28845847066340830932319815946, 5.80535536640418513374154621683, 6.71324692677754984647032212019, 7.55669562500910880174645327086, 8.238233458105091746262229727887, 9.219246571923555088756076227604, 10.00282988693479946451534781656

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