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Properties

Label 2-1008-252.115-c1-0-44
Degree $2$
Conductor $1008$
Sign $-0.991 - 0.132i$
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  + (1.29 − 1.15i)3-s + (−2.47 − 1.43i)5-s + (1.38 + 2.25i)7-s + (0.340 − 2.98i)9-s + (−3.57 + 2.06i)11-s + (−3.14 + 1.81i)13-s + (−4.85 + 1.00i)15-s + (−3.36 − 1.94i)17-s + (−3.57 − 6.18i)19-s + (4.39 + 1.30i)21-s + (−5.18 − 2.99i)23-s + (1.60 + 2.77i)25-s + (−2.99 − 4.24i)27-s + (−2.87 + 4.98i)29-s + 10.4·31-s + ⋯
L(s)  = 1  + (0.746 − 0.665i)3-s + (−1.10 − 0.640i)5-s + (0.524 + 0.851i)7-s + (0.113 − 0.993i)9-s + (−1.07 + 0.622i)11-s + (−0.870 + 0.502i)13-s + (−1.25 + 0.260i)15-s + (−0.815 − 0.470i)17-s + (−0.819 − 1.41i)19-s + (0.958 + 0.285i)21-s + (−1.08 − 0.623i)23-s + (0.320 + 0.554i)25-s + (−0.576 − 0.816i)27-s + (−0.534 + 0.925i)29-s + 1.88·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.991 - 0.132i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4262380619\)
\(L(\frac12)\) \(\approx\) \(0.4262380619\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.15i)T \)
7 \( 1 + (-1.38 - 2.25i)T \)
good5 \( 1 + (2.47 + 1.43i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.57 - 2.06i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.14 - 1.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.36 + 1.94i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.57 + 6.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.18 + 2.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.87 - 4.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (2.02 + 3.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.64 + 1.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.533 - 0.308i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.07T + 47T^{2} \)
53 \( 1 + (2.65 - 4.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.29T + 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 + 1.58iT - 67T^{2} \)
71 \( 1 + 5.90iT - 71T^{2} \)
73 \( 1 + (6.78 + 3.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.05iT - 79T^{2} \)
83 \( 1 + (-1.69 + 2.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9.80 + 5.65i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.52 - 1.45i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.056994229743324660215747171437, −8.755661728986052446697267678593, −7.82281178299074514985616722955, −7.34896405155386821394972566980, −6.28865719488646854433758301200, −4.77225141091030989774687271981, −4.45376937347066083441740215273, −2.77430028794783169874158400586, −2.09317102426809609285098334993, −0.16203710110148259134067419513, 2.21165324598326890623216885244, 3.35022225630313255640876209945, 4.07332751016796319680704393474, 4.86242079171031427879513146869, 6.17195055399921337620541680830, 7.49259882816693630540186321058, 8.028925370824757244605465705025, 8.292693948285934283149851708453, 9.882362280157253269154738468913, 10.32759955400506840038478506499

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