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Properties

Label 2-1008-252.103-c1-0-43
Degree $2$
Conductor $1008$
Sign $-0.992 - 0.125i$
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  + (−1.60 − 0.656i)3-s + (2.94 − 1.70i)5-s + (−1.13 + 2.38i)7-s + (2.13 + 2.10i)9-s + (−4.77 − 2.75i)11-s + (−1.96 − 1.13i)13-s + (−5.84 + 0.791i)15-s + (−6.64 + 3.83i)17-s + (0.850 − 1.47i)19-s + (3.38 − 3.08i)21-s + (−1.09 + 0.634i)23-s + (3.29 − 5.70i)25-s + (−2.04 − 4.77i)27-s + (−3.04 − 5.26i)29-s + 7.32·31-s + ⋯
L(s)  = 1  + (−0.925 − 0.379i)3-s + (1.31 − 0.761i)5-s + (−0.429 + 0.903i)7-s + (0.712 + 0.701i)9-s + (−1.43 − 0.830i)11-s + (−0.545 − 0.314i)13-s + (−1.50 + 0.204i)15-s + (−1.61 + 0.931i)17-s + (0.195 − 0.338i)19-s + (0.739 − 0.673i)21-s + (−0.229 + 0.132i)23-s + (0.658 − 1.14i)25-s + (−0.393 − 0.919i)27-s + (−0.564 − 0.978i)29-s + 1.31·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.992 - 0.125i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2235731726\)
\(L(\frac12)\) \(\approx\) \(0.2235731726\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.60 + 0.656i)T \)
7 \( 1 + (1.13 - 2.38i)T \)
good5 \( 1 + (-2.94 + 1.70i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.77 + 2.75i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.96 + 1.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.64 - 3.83i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.850 + 1.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 - 0.634i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.04 + 5.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.32T + 31T^{2} \)
37 \( 1 + (0.928 - 1.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.99 + 2.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.68 - 3.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.64T + 47T^{2} \)
53 \( 1 + (2.90 + 5.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.97T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + 8.24iT - 67T^{2} \)
71 \( 1 + 5.47iT - 71T^{2} \)
73 \( 1 + (12.5 - 7.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.42iT - 79T^{2} \)
83 \( 1 + (-5.67 - 9.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.67 - 2.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.903 + 0.521i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.701055841888111885267635915582, −8.679104397569156268838303622377, −8.016354164827962295004359575169, −6.60954540265838063092650783609, −5.98476152212890302270216019638, −5.37500768205331094731514916726, −4.67559128004192837431710608859, −2.70036946779576420778405308766, −1.84969562698564775670077640830, −0.10309245427150370586409655247, 1.93922524736731325366693506025, 3.04983470157326193258610264164, 4.57254775700716708226148752893, 5.12525021413726206461514719329, 6.27119716590794956384885178736, 6.84575220992399982929952375040, 7.50925143128477284660017724653, 9.143747993127985069756990165700, 9.941702415921494446008867805279, 10.28210168409851005662799870187

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