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Properties

Label 2-1008-63.58-c1-0-41
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: $\chi_{1008} (625, \cdot )$
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

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

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