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Properties

Label 2-1008-63.58-c1-0-31
Degree $2$
Conductor $1008$
Sign $-0.580 + 0.814i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + (−0.5 − 0.866i)5-s + (−2 + 1.73i)7-s − 2.99·9-s + (−1.5 + 2.59i)11-s + (−0.5 + 0.866i)13-s + (1.49 − 0.866i)15-s + (−1.5 − 2.59i)17-s + (2.5 − 4.33i)19-s + (−2.99 − 3.46i)21-s + (0.5 + 0.866i)23-s + (2 − 3.46i)25-s − 5.19i·27-s + (−4.5 − 7.79i)29-s − 4·31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s − 0.999·9-s + (−0.452 + 0.783i)11-s + (−0.138 + 0.240i)13-s + (0.387 − 0.223i)15-s + (−0.363 − 0.630i)17-s + (0.573 − 0.993i)19-s + (−0.654 − 0.755i)21-s + (0.104 + 0.180i)23-s + (0.400 − 0.692i)25-s − 0.999i·27-s + (−0.835 − 1.44i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.580 + 0.814i$
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: \(1\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.5 - 2.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-6.5 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (6.5 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.640813084017751884046979873691, −9.082907810347376959998651048511, −8.225002472532366261414108130122, −7.13374651509719037938873481749, −6.12914338337346937117504245595, −5.08296974785184007746066599780, −4.54339003012016005954261696100, −3.30769056500662343019213428366, −2.38635645349822569884973618318, 0, 1.56125568281334883019518887480, 3.06459731335624296264540224880, 3.65707512955832332916819981104, 5.35878937451834078505323644327, 6.08625612207211162835708968721, 7.07852226127769372973605577338, 7.50533697208485644202273568578, 8.489639522096411240271802810640, 9.293786222245745217116822390839

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