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Properties

Label 2-1008-63.25-c1-0-17
Degree $2$
Conductor $1008$
Sign $-0.0574 - 0.998i$
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  + (1.69 + 0.332i)3-s + (−1.59 + 2.76i)5-s + (1.66 + 2.05i)7-s + (2.77 + 1.12i)9-s + (−1.14 − 1.97i)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 + (2.15 + 4.04i)21-s + (−3.23 + 5.60i)23-s + (−2.60 − 4.51i)25-s + (4.34 + 2.84i)27-s + (−1.06 + 1.83i)29-s + 0.632·31-s + ⋯
L(s)  = 1  + (0.981 + 0.191i)3-s + (−0.714 + 1.23i)5-s + (0.629 + 0.776i)7-s + (0.926 + 0.376i)9-s + (−0.344 − 0.596i)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 + (0.469 + 0.883i)21-s + (−0.674 + 1.16i)23-s + (−0.520 − 0.902i)25-s + (0.837 + 0.547i)27-s + (−0.197 + 0.341i)29-s + 0.113·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0574 - 0.998i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.0574 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033678622\)
\(L(\frac12)\) \(\approx\) \(2.033678622\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.332i)T \)
7 \( 1 + (-1.66 - 2.05i)T \)
good5 \( 1 + (1.59 - 2.76i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.14 + 1.97i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (3.23 - 5.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.06 - 1.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.632T + 31T^{2} \)
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 \( 1 + 6.53T + 47T^{2} \)
53 \( 1 + (-2.39 + 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.20T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 - 3.01T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 + 7.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.87T + 79T^{2} \)
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

−10.06616987964224227112092655025, −9.441667218654535316798596396642, −8.202128835647925832485439389515, −7.87252161726619161759215892333, −7.14675942508003136036209629919, −5.86745179982922955644128292717, −4.91186371067303000899330758247, −3.40497520312253890804477835334, −3.19486793638172944152676818394, −1.86266260034853682977401423414, 0.869072002241613226822329350331, 2.07596465106073520184173056965, 3.55458528287124270681612402192, 4.48116054567564030071501381291, 4.94524670891139407070040944567, 6.62954646755884474451533035713, 7.58923733191292401561758277552, 8.060286362234581427741161545259, 8.746472934171418641613179820813, 9.606113356465428650207048450710

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