Properties

Label 2-1008-252.115-c1-0-10
Degree $2$
Conductor $1008$
Sign $0.955 - 0.293i$
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  + (−1.42 + 0.987i)3-s + (−0.679 − 0.392i)5-s + (−2.49 − 0.891i)7-s + (1.04 − 2.81i)9-s + (−1.22 + 0.708i)11-s + (−1.50 + 0.868i)13-s + (1.35 − 0.112i)15-s + (5.43 + 3.13i)17-s + (0.736 + 1.27i)19-s + (4.42 − 1.19i)21-s + (−4.85 − 2.80i)23-s + (−2.19 − 3.79i)25-s + (1.28 + 5.03i)27-s + (3.95 − 6.85i)29-s + 8.41·31-s + ⋯
L(s)  = 1  + (−0.821 + 0.570i)3-s + (−0.303 − 0.175i)5-s + (−0.941 − 0.337i)7-s + (0.349 − 0.936i)9-s + (−0.369 + 0.213i)11-s + (−0.417 + 0.240i)13-s + (0.349 − 0.0291i)15-s + (1.31 + 0.761i)17-s + (0.168 + 0.292i)19-s + (0.965 − 0.259i)21-s + (−1.01 − 0.584i)23-s + (−0.438 − 0.759i)25-s + (0.246 + 0.969i)27-s + (0.734 − 1.27i)29-s + 1.51·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.955 - 0.293i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.955 - 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8868395809\)
\(L(\frac12)\) \(\approx\) \(0.8868395809\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 - 0.987i)T \)
7 \( 1 + (2.49 + 0.891i)T \)
good5 \( 1 + (0.679 + 0.392i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.22 - 0.708i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.50 - 0.868i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.43 - 3.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.736 - 1.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.85 + 2.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.95 + 6.85i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.41T + 31T^{2} \)
37 \( 1 + (-3.74 - 6.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.19 + 4.15i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.85 - 4.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.110T + 47T^{2} \)
53 \( 1 + (4.28 - 7.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.0736T + 59T^{2} \)
61 \( 1 + 1.23iT - 61T^{2} \)
67 \( 1 - 11.8iT - 67T^{2} \)
71 \( 1 + 0.390iT - 71T^{2} \)
73 \( 1 + (-3.70 - 2.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.00iT - 79T^{2} \)
83 \( 1 + (-7.88 + 13.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.15 + 3.55i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.89 - 3.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.03421432423860410247501645211, −9.592398436328969752746109645036, −8.256732183541885237222039066951, −7.51896365630739432483557493328, −6.24240265740127936153522344813, −5.96663991323598717685861347823, −4.55763248510749231480551194104, −4.02007942214656696375496435621, −2.77382936747975833338098947966, −0.75292591998861548449796846618, 0.74945409087265043503886640068, 2.48821227394352426141027910778, 3.50299803290762548727869389664, 4.94129336210880328131693941612, 5.70851224254883851863151140036, 6.46176625097283681399236946609, 7.46642497833345745920262758119, 7.898992887863212510684737059507, 9.279936966195200120940059333765, 9.985022821893899769779655508553

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