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Properties

Label 2-1008-63.38-c1-0-30
Degree $2$
Conductor $1008$
Sign $0.880 + 0.474i$
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  + (1.68 − 0.397i)3-s + (−0.311 + 0.540i)5-s + (−2.62 + 0.362i)7-s + (2.68 − 1.34i)9-s + (4.50 − 2.59i)11-s + (2.74 − 1.58i)13-s + (−0.310 + 1.03i)15-s + (−0.437 + 0.757i)17-s + (1.41 − 0.819i)19-s + (−4.27 + 1.65i)21-s + (−7.14 − 4.12i)23-s + (2.30 + 3.99i)25-s + (3.99 − 3.32i)27-s + (4.96 + 2.86i)29-s − 4.64i·31-s + ⋯
L(s)  = 1  + (0.973 − 0.229i)3-s + (−0.139 + 0.241i)5-s + (−0.990 + 0.136i)7-s + (0.894 − 0.446i)9-s + (1.35 − 0.783i)11-s + (0.760 − 0.438i)13-s + (−0.0802 + 0.267i)15-s + (−0.106 + 0.183i)17-s + (0.325 − 0.187i)19-s + (−0.932 + 0.360i)21-s + (−1.49 − 0.860i)23-s + (0.461 + 0.798i)25-s + (0.768 − 0.640i)27-s + (0.921 + 0.532i)29-s − 0.834i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.214346983\)
\(L(\frac12)\) \(\approx\) \(2.214346983\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.397i)T \)
7 \( 1 + (2.62 - 0.362i)T \)
good5 \( 1 + (0.311 - 0.540i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.50 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.74 + 1.58i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.437 - 0.757i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 0.819i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.14 + 4.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.96 - 2.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
37 \( 1 + (-1.24 - 2.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.52 - 6.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.56 + 2.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.46T + 47T^{2} \)
53 \( 1 + (-1.15 - 0.665i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.36T + 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (11.6 + 6.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 + (0.192 - 0.332i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.0198 + 0.0344i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.94 + 3.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.753588640128056201065294795522, −8.938281462448757286073958143611, −8.451511052821521308364298922720, −7.40106190365028865295407277866, −6.48049218210051802030435137636, −5.97640393684624452744535178272, −4.18745826065282473496609346858, −3.51176114633917516736498455530, −2.65675025106718890352508945276, −1.09030792439100135256072728144, 1.41132013372558700335424580137, 2.72584448698966395513330774523, 3.97301202762802954918288362871, 4.22475256841321624096696217602, 5.88305199499536066915117954821, 6.78880083562263334123449170759, 7.50194704457060317068406131325, 8.627894950739530417723969079012, 9.152535929707226705441536643376, 9.871192188197853759956909984153

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