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Properties

Label 2-1008-252.31-c1-0-25
Degree $2$
Conductor $1008$
Sign $-0.237 + 0.971i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (−1.13 − 1.30i)3-s + 0.618i·5-s + (−1.03 + 2.43i)7-s + (−0.413 + 2.97i)9-s − 2.97i·11-s + (−1.64 − 0.951i)13-s + (0.808 − 0.703i)15-s + (2.92 + 1.68i)17-s + (−1.13 − 1.96i)19-s + (4.35 − 1.41i)21-s − 6.74i·23-s + 4.61·25-s + (4.35 − 2.83i)27-s + (−1.74 − 3.02i)29-s + (−3.67 − 6.36i)31-s + ⋯
L(s)  = 1  + (−0.656 − 0.754i)3-s + 0.276i·5-s + (−0.391 + 0.920i)7-s + (−0.137 + 0.990i)9-s − 0.898i·11-s + (−0.457 − 0.263i)13-s + (0.208 − 0.181i)15-s + (0.708 + 0.409i)17-s + (−0.260 − 0.450i)19-s + (0.951 − 0.308i)21-s − 1.40i·23-s + 0.923·25-s + (0.837 − 0.546i)27-s + (−0.324 − 0.562i)29-s + (−0.660 − 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.237 + 0.971i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.237 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8516370963\)
\(L(\frac12)\) \(\approx\) \(0.8516370963\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 + 1.30i)T \)
7 \( 1 + (1.03 - 2.43i)T \)
good5 \( 1 - 0.618iT - 5T^{2} \)
11 \( 1 + 2.97iT - 11T^{2} \)
13 \( 1 + (1.64 + 0.951i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.92 - 1.68i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.13 + 1.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.74iT - 23T^{2} \)
29 \( 1 + (1.74 + 3.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.67 + 6.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.25 + 5.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.09 - 5.25i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.48 + 1.43i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.80 - 4.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.50 + 4.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.17 + 3.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.14 + 1.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.34 + 2.50i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.18iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.4 + 6.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.48 - 9.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.03 - 3.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.71 + 1.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.760980357359612092834956600834, −8.779865211101115488692122658285, −8.016060105462993829672844857519, −7.12145814666132592164132892046, −6.11102966274590235527639514067, −5.77309476231535146334298928243, −4.62273851694648273997721193552, −3.08674831167058827155661271105, −2.19132778657272623973792706824, −0.45989482111831130252755367335, 1.28425876749259158749987965957, 3.18990629842939881525548400434, 4.11267972721731013221700053053, 4.94225173558519656638907289452, 5.73975034031262058799472698043, 6.98172482152686333679316070612, 7.38814156177014925003993075122, 8.816328994828496896796097416919, 9.584245439510274401065406833061, 10.17839772119571646245435684764

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