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Properties

Label 2-1008-84.83-c2-0-26
Degree $2$
Conductor $1008$
Sign $0.000539 + 0.999i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.34·5-s + (−0.646 − 6.97i)7-s + 7.76·11-s − 13.4i·13-s − 28.9·17-s + 32.7·19-s + 16.8·23-s − 6.12·25-s − 38.3i·29-s − 50.1·31-s + (−2.80 − 30.2i)35-s − 39.1·37-s + 63.0·41-s − 7.01i·43-s + 23.4i·47-s + ⋯
L(s)  = 1  + 0.868·5-s + (−0.0922 − 0.995i)7-s + 0.705·11-s − 1.03i·13-s − 1.70·17-s + 1.72·19-s + 0.732·23-s − 0.245·25-s − 1.32i·29-s − 1.61·31-s + (−0.0801 − 0.865i)35-s − 1.05·37-s + 1.53·41-s − 0.163i·43-s + 0.499i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000539 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.000539 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.000539 + 0.999i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.000539 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.978908641\)
\(L(\frac12)\) \(\approx\) \(1.978908641\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (0.646 + 6.97i)T \)
good5 \( 1 - 4.34T + 25T^{2} \)
11 \( 1 - 7.76T + 121T^{2} \)
13 \( 1 + 13.4iT - 169T^{2} \)
17 \( 1 + 28.9T + 289T^{2} \)
19 \( 1 - 32.7T + 361T^{2} \)
23 \( 1 - 16.8T + 529T^{2} \)
29 \( 1 + 38.3iT - 841T^{2} \)
31 \( 1 + 50.1T + 961T^{2} \)
37 \( 1 + 39.1T + 1.36e3T^{2} \)
41 \( 1 - 63.0T + 1.68e3T^{2} \)
43 \( 1 + 7.01iT - 1.84e3T^{2} \)
47 \( 1 - 23.4iT - 2.20e3T^{2} \)
53 \( 1 + 90.9iT - 2.80e3T^{2} \)
59 \( 1 - 78.1iT - 3.48e3T^{2} \)
61 \( 1 - 49.9iT - 3.72e3T^{2} \)
67 \( 1 + 15.8iT - 4.48e3T^{2} \)
71 \( 1 - 62.1T + 5.04e3T^{2} \)
73 \( 1 + 59.7iT - 5.32e3T^{2} \)
79 \( 1 + 137. iT - 6.24e3T^{2} \)
83 \( 1 + 102. iT - 6.88e3T^{2} \)
89 \( 1 + 26.6T + 7.92e3T^{2} \)
97 \( 1 - 105. iT - 9.40e3T^{2} \)
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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

−9.524246424916973538161892504333, −8.991388582545590900764723203087, −7.72249140284835557692854487256, −7.07106734033665296240715727753, −6.13691741983326285433222328923, −5.29477627784324177852011687499, −4.20857996172637759878995373083, −3.19782091050711932750829632349, −1.88398049662797412232911343229, −0.61532650272197707226296661734, 1.52739402215752635175502489646, 2.40825197772200581088919360620, 3.65560898329031902151399889014, 4.92790444111841003617203924315, 5.66561356480232452723985232019, 6.59533540899238264836578342411, 7.23079658037217843741677476044, 8.695692422766840111595585850334, 9.251783642226347687230858985824, 9.571583613005449228873214576770

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