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Properties

Label 2-1008-63.16-c1-0-18
Degree $2$
Conductor $1008$
Sign $0.924 - 0.382i$
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  + (−1.13 + 1.30i)3-s + 3.19·5-s + (−2.61 − 0.415i)7-s + (−0.411 − 2.97i)9-s + 2.28·11-s + (−0.675 − 1.16i)13-s + (−3.63 + 4.17i)15-s + (2.21 + 3.83i)17-s + (3.69 − 6.39i)19-s + (3.51 − 2.93i)21-s + 6.46·23-s + 5.20·25-s + (4.34 + 2.84i)27-s + (−1.06 + 1.83i)29-s + (−0.316 + 0.547i)31-s + ⋯
L(s)  = 1  + (−0.656 + 0.754i)3-s + 1.42·5-s + (−0.987 − 0.157i)7-s + (−0.137 − 0.990i)9-s + 0.688·11-s + (−0.187 − 0.324i)13-s + (−0.938 + 1.07i)15-s + (0.537 + 0.930i)17-s + (0.847 − 1.46i)19-s + (0.767 − 0.641i)21-s + 1.34·23-s + 1.04·25-s + (0.837 + 0.547i)27-s + (−0.197 + 0.341i)29-s + (−0.0567 + 0.0983i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.924 - 0.382i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.924 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580937307\)
\(L(\frac12)\) \(\approx\) \(1.580937307\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 1.30i)T \)
7 \( 1 + (2.61 + 0.415i)T \)
good5 \( 1 - 3.19T + 5T^{2} \)
11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + (0.675 + 1.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.69 + 6.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.46T + 23T^{2} \)
29 \( 1 + (1.06 - 1.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.316 - 0.547i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.92 + 3.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.05 + 8.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.24 - 7.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.26 - 5.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.39 - 4.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.10 + 5.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.45 - 7.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.50 - 2.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 - 7.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.938 + 1.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.00 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.65 + 4.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.44 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.959170441991473602448779556271, −9.358399735164427773973935753851, −8.875188679526848831581578314477, −7.11988040534835771806056680487, −6.47679016828957552721729138279, −5.67646850428291207048475997655, −5.02646959267910735365838233839, −3.72626837008319817470439110569, −2.76478042742726519519947039961, −1.03954201323631115332755070653, 1.10660322221543248908992141868, 2.22898953225087879241966436041, 3.40043995269915666031581878106, 5.12582831526938520987849813101, 5.67245705911333796471361527726, 6.56731278149667993630591476333, 6.99551714607338160111778837127, 8.212576914182894780030608065322, 9.478076703310900808992742334338, 9.708345483608305798426964737011

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