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Properties

Label 2-1008-252.103-c1-0-39
Degree $2$
Conductor $1008$
Sign $-0.313 + 0.949i$
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  + (−1.62 + 0.608i)3-s + (1.53 − 0.886i)5-s + (2.64 + 0.0987i)7-s + (2.25 − 1.97i)9-s + (−5.33 − 3.08i)11-s + (−4.60 − 2.65i)13-s + (−1.95 + 2.37i)15-s + (4.69 − 2.71i)17-s + (−0.935 + 1.62i)19-s + (−4.34 + 1.44i)21-s + (0.562 − 0.324i)23-s + (−0.928 + 1.60i)25-s + (−2.46 + 4.57i)27-s + (−1.14 − 1.98i)29-s − 10.2·31-s + ⋯
L(s)  = 1  + (−0.936 + 0.351i)3-s + (0.686 − 0.396i)5-s + (0.999 + 0.0373i)7-s + (0.752 − 0.658i)9-s + (−1.60 − 0.929i)11-s + (−1.27 − 0.737i)13-s + (−0.503 + 0.612i)15-s + (1.13 − 0.657i)17-s + (−0.214 + 0.371i)19-s + (−0.948 + 0.316i)21-s + (0.117 − 0.0677i)23-s + (−0.185 + 0.321i)25-s + (−0.473 + 0.880i)27-s + (−0.213 − 0.369i)29-s − 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.313 + 0.949i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8527838300\)
\(L(\frac12)\) \(\approx\) \(0.8527838300\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.608i)T \)
7 \( 1 + (-2.64 - 0.0987i)T \)
good5 \( 1 + (-1.53 + 0.886i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.33 + 3.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.60 + 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.69 + 2.71i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.935 - 1.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.562 + 0.324i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.14 + 1.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.64 + 3.83i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.07 + 0.620i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.468T + 47T^{2} \)
53 \( 1 + (0.941 + 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.24T + 59T^{2} \)
61 \( 1 + 6.33iT - 61T^{2} \)
67 \( 1 + 9.93iT - 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (4.77 - 2.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 + (4.06 + 7.03i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.228 + 0.132i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.5 - 7.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.952080300501676576852289874223, −9.030457637032209562473573807465, −7.85624294060979457306271861974, −7.39296880336702335198996135550, −5.81413228252294321575241076301, −5.31920979409725379669615562294, −4.95437417866844460501952014719, −3.38762827186232428012804179746, −1.99923949516808514644931638516, −0.41743909150968378715045712089, 1.70335164173498026031872851046, 2.46346961804501473914969598778, 4.37959448911004064944856706173, 5.21576561784420018058964270903, 5.68723829673650045231346310582, 7.07523025184474424691503795044, 7.39589445272769312007755359606, 8.358916983007458445368730496322, 9.856484218322635501176168296339, 10.15267359427920217218751382229

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