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Properties

Label 2-1008-9.4-c1-0-28
Degree $2$
Conductor $1008$
Sign $0.787 + 0.616i$
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  + (1.71 + 0.272i)3-s + (−0.119 + 0.207i)5-s + (−0.5 − 0.866i)7-s + (2.85 + 0.931i)9-s + (−2.56 − 4.43i)11-s + (2.44 − 4.23i)13-s + (−0.260 + 0.321i)15-s + 3.70·17-s − 3.66·19-s + (−0.619 − 1.61i)21-s + (3.71 − 6.42i)23-s + (2.47 + 4.28i)25-s + (4.62 + 2.36i)27-s + (−1.73 − 3.00i)29-s + (−0.358 + 0.621i)31-s + ⋯
L(s)  = 1  + (0.987 + 0.157i)3-s + (−0.0534 + 0.0926i)5-s + (−0.188 − 0.327i)7-s + (0.950 + 0.310i)9-s + (−0.772 − 1.33i)11-s + (0.677 − 1.17i)13-s + (−0.0673 + 0.0830i)15-s + 0.898·17-s − 0.839·19-s + (−0.135 − 0.352i)21-s + (0.773 − 1.34i)23-s + (0.494 + 0.856i)25-s + (0.890 + 0.455i)27-s + (−0.321 − 0.557i)29-s + (−0.0644 + 0.111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.196369977\)
\(L(\frac12)\) \(\approx\) \(2.196369977\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.272i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.119 - 0.207i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.56 + 4.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 4.23i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + (-3.71 + 6.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.73 + 3.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.358 - 0.621i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.60T + 37T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.24 - 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.16 - 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.942T + 53T^{2} \)
59 \( 1 + (-3.78 + 6.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.75 + 4.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.330 - 0.571i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 3.66T + 73T^{2} \)
79 \( 1 + (3.11 + 5.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.85 - 8.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 + (-8.57 - 14.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.875416412019438901587905933247, −8.916652987954882668304751577397, −8.148220579321317417429000342434, −7.74235781435002959507129470103, −6.49864216357453922929306572848, −5.58615581834618453136029034650, −4.42835323597690597869210036442, −3.26166044699315201512058130211, −2.81539813650829824982620371699, −0.977744371199471365855692125168, 1.62105816068170743284440977895, 2.56424208696084448468478371086, 3.75576054487435265407405397482, 4.60779012261765606414926685487, 5.77652674658637086027353784973, 7.04014187990760149975857624584, 7.42168436179365299033962566778, 8.574453292577486638937386717073, 9.081202660317039716187030127923, 9.930199249676000893156041074650

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