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Properties

Label 2-1008-63.4-c1-0-19
Degree $2$
Conductor $1008$
Sign $0.945 + 0.325i$
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 + 1.58·5-s + (2.64 + 0.0963i)7-s + (0.349 + 2.97i)9-s + 1.58·11-s + (2.40 − 4.16i)13-s + (−2.05 − 1.82i)15-s + (−2.69 + 4.67i)17-s + (3.54 + 6.14i)19-s + (−3.31 − 3.16i)21-s − 0.300·23-s − 2.47·25-s + (2.97 − 4.25i)27-s + (4.13 + 7.16i)29-s + (−1.35 − 2.34i)31-s + ⋯
L(s)  = 1  + (−0.747 − 0.664i)3-s + 0.710·5-s + (0.999 + 0.0364i)7-s + (0.116 + 0.993i)9-s + 0.478·11-s + (0.667 − 1.15i)13-s + (−0.530 − 0.472i)15-s + (−0.654 + 1.13i)17-s + (0.814 + 1.41i)19-s + (−0.722 − 0.691i)21-s − 0.0626·23-s − 0.495·25-s + (0.572 − 0.819i)27-s + (0.768 + 1.33i)29-s + (−0.243 − 0.421i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.945 + 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.675755737\)
\(L(\frac12)\) \(\approx\) \(1.675755737\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.0963i)T \)
good5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 1.58T + 11T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 - 4.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.54 - 6.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.300T + 23T^{2} \)
29 \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.35 + 2.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.833 - 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 + 2.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.23 - 5.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.23 + 3.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.02 + 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 + 13.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.19 + 7.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.18 + 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.60 - 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.34172452314358318725386424921, −8.952685463622286744520086635429, −8.122984171972419401304774348328, −7.48501929437643842840412554001, −6.24518668740519708190786609549, −5.79069348042530900973526298359, −4.95514572151598576087960067765, −3.67426835935735837502067876534, −2.01720711100601881349038351943, −1.22166088279251945802949777836, 1.09739730595385598535311266963, 2.51252440684941007060492351017, 4.12178166477437790324780144130, 4.73031098073847314139868641832, 5.62744788924726695899493397715, 6.51871537888266964889641056207, 7.27115012664700137152391317463, 8.675432962954031932116098866256, 9.292390092876109516648246530816, 9.910140797316017713160943823936

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