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Properties

Label 2-1008-7.3-c2-0-37
Degree $2$
Conductor $1008$
Sign $-0.984 + 0.174i$
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  + (6.80 − 3.92i)5-s + (−6.99 + 0.337i)7-s + (−10.2 + 17.7i)11-s + 4.69i·13-s + (−15.8 − 9.16i)17-s + (16.4 − 9.48i)19-s + (−14.6 − 25.3i)23-s + (18.3 − 31.8i)25-s − 37.7·29-s + (−29.4 − 17.0i)31-s + (−46.2 + 29.7i)35-s + (−21.4 − 37.1i)37-s − 25.4i·41-s − 10.5·43-s + (8.64 − 4.99i)47-s + ⋯
L(s)  = 1  + (1.36 − 0.785i)5-s + (−0.998 + 0.0481i)7-s + (−0.929 + 1.60i)11-s + 0.360i·13-s + (−0.934 − 0.539i)17-s + (0.864 − 0.499i)19-s + (−0.637 − 1.10i)23-s + (0.735 − 1.27i)25-s − 1.30·29-s + (−0.949 − 0.548i)31-s + (−1.32 + 0.850i)35-s + (−0.578 − 1.00i)37-s − 0.620i·41-s − 0.245·43-s + (0.183 − 0.106i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.984 + 0.174i$
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: \((2,\ 1008,\ (\ :1),\ -0.984 + 0.174i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3803360636\)
\(L(\frac12)\) \(\approx\) \(0.3803360636\)
\(L(2)\) 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 \( 1 + (6.99 - 0.337i)T \)
good5 \( 1 + (-6.80 + 3.92i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (10.2 - 17.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 4.69iT - 169T^{2} \)
17 \( 1 + (15.8 + 9.16i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-16.4 + 9.48i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (14.6 + 25.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 37.7T + 841T^{2} \)
31 \( 1 + (29.4 + 17.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (21.4 + 37.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 25.4iT - 1.68e3T^{2} \)
43 \( 1 + 10.5T + 1.84e3T^{2} \)
47 \( 1 + (-8.64 + 4.99i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (42.0 - 72.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (12.3 + 7.15i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-41.7 + 24.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.9 - 51.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 111.T + 5.04e3T^{2} \)
73 \( 1 + (-34.5 - 19.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (42.2 + 73.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 41.7iT - 6.88e3T^{2} \)
89 \( 1 + (-4.10 + 2.36i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 138. iT - 9.40e3T^{2} \)
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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

−9.314402775373027188615947735285, −9.024842251226479759733772045938, −7.52459189706650808245782192649, −6.84600254133224340152943505528, −5.82138769828377212507593795755, −5.11936252488108947314740157652, −4.19814653363287267130753720594, −2.56098539710217605876652526568, −1.89691295242230010193856493922, −0.10451580833677380514097059015, 1.74358255350086400267907465558, 2.98538607823704218021255119548, 3.51167505485884955575044376069, 5.43139631934465884537781629001, 5.84265576473925703460358406718, 6.58187438936931820170423740489, 7.57793281182302933919638232920, 8.626501338378530935592827737352, 9.524194877828431301671198702381, 10.12883804909854141657167302099

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