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Properties

Label 2-1008-252.103-c1-0-31
Degree $2$
Conductor $1008$
Sign $0.977 + 0.209i$
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  + (1.69 − 0.331i)3-s + (−0.535 + 0.309i)5-s + (2.62 + 0.319i)7-s + (2.77 − 1.12i)9-s + (−2.58 − 1.48i)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 + (4.57 − 0.328i)21-s + (5.84 − 3.37i)23-s + (−2.30 + 3.99i)25-s + (4.35 − 2.83i)27-s + (−1.74 − 3.02i)29-s + 7.35·31-s + ⋯
L(s)  = 1  + (0.981 − 0.191i)3-s + (−0.239 + 0.138i)5-s + (0.992 + 0.120i)7-s + (0.926 − 0.376i)9-s + (−0.778 − 0.449i)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 + (0.997 − 0.0715i)21-s + (1.21 − 0.703i)23-s + (−0.461 + 0.799i)25-s + (0.837 − 0.546i)27-s + (−0.324 − 0.562i)29-s + 1.32·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.977 + 0.209i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.977 + 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.459503075\)
\(L(\frac12)\) \(\approx\) \(2.459503075\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.331i)T \)
7 \( 1 + (-2.62 - 0.319i)T \)
good5 \( 1 + (0.535 - 0.309i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.58 + 1.48i)T + (5.5 + 9.52i)T^{2} \)
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 \( 1 + (-5.84 + 3.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.74 + 3.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.35T + 31T^{2} \)
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 \( 1 - 5.60T + 47T^{2} \)
53 \( 1 + (-2.50 - 4.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 + 3.63iT - 61T^{2} \)
67 \( 1 + 5.01iT - 67T^{2} \)
71 \( 1 - 6.18iT - 71T^{2} \)
73 \( 1 + (12.5 - 7.23i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
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

−9.937523381633319756799720968871, −8.815659821250885739827483211666, −8.317107785000626603381442405148, −7.63007484223120978260200927034, −6.79565095211819744461675417736, −5.53572302668483990416887437872, −4.58003347683487343978002427781, −3.49427826067878787544908084097, −2.54947511997340766615606365775, −1.29513666449727266818600821627, 1.40362949702385750266417738323, 2.60373851845623234812863117845, 3.69281159447544233111973307803, 4.67174459680908993020849822861, 5.41489786768391179719856490992, 6.91833247573579691643355648921, 7.71550664668828259466465075930, 8.313161251556714281846827527750, 8.955806580686654971465632172964, 10.08952752531731719812772368769

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