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Properties

Label 2-1008-63.16-c1-0-12
Degree $2$
Conductor $1008$
Sign $0.194 - 0.980i$
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  + (1.72 + 0.133i)3-s − 4.22·5-s + (2.37 + 1.15i)7-s + (2.96 + 0.460i)9-s − 1.92·11-s + (−0.291 − 0.504i)13-s + (−7.29 − 0.562i)15-s + (3.61 + 6.25i)17-s + (−2.10 + 3.64i)19-s + (3.95 + 2.31i)21-s − 1.27·23-s + 12.8·25-s + (5.05 + 1.18i)27-s + (−4.20 + 7.27i)29-s + (−0.476 + 0.824i)31-s + ⋯
L(s)  = 1  + (0.997 + 0.0769i)3-s − 1.88·5-s + (0.898 + 0.438i)7-s + (0.988 + 0.153i)9-s − 0.581·11-s + (−0.0808 − 0.140i)13-s + (−1.88 − 0.145i)15-s + (0.875 + 1.51i)17-s + (−0.482 + 0.835i)19-s + (0.862 + 0.506i)21-s − 0.266·23-s + 2.56·25-s + (0.973 + 0.228i)27-s + (−0.780 + 1.35i)29-s + (−0.0855 + 0.148i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.194 - 0.980i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.549345356\)
\(L(\frac12)\) \(\approx\) \(1.549345356\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.133i)T \)
7 \( 1 + (-2.37 - 1.15i)T \)
good5 \( 1 + 4.22T + 5T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 + (0.291 + 0.504i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.61 - 6.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.10 - 3.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.27T + 23T^{2} \)
29 \( 1 + (4.20 - 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.476 - 0.824i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.03 + 5.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.31 - 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.442 - 0.766i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.88 - 4.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.962 + 1.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.27 - 3.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.43 - 4.21i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-0.446 - 0.772i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.93 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.24 + 9.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.87 + 6.71i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.98 - 3.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.33861024412767054321709232493, −8.900873905101125929884483673710, −8.385071623493811879421329022501, −7.74423794216638466015357307808, −7.35968867849600876190475157390, −5.76556770250772133790784391992, −4.54458836533271786958605148962, −3.88353892028070845517458888555, −3.02421321870581285726587370387, −1.57167951023214223261118066732, 0.66805420763383641781291574207, 2.46802466886841608363819003924, 3.51445141611494966149299330619, 4.34040941612806404120105030543, 5.02087132071057525070059448328, 6.89232810530766841707686584090, 7.64496172115436084309528535070, 7.88197000686803735369166522813, 8.706214554861864399984397862201, 9.666362207143592102694308930613

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