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Properties

Label 2-1008-252.31-c1-0-47
Degree $2$
Conductor $1008$
Sign $-0.471 - 0.882i$
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  + (−0.232 − 1.71i)3-s − 3.40i·5-s + (−2.63 − 0.211i)7-s + (−2.89 + 0.798i)9-s + 5.50i·11-s + (1.96 + 1.13i)13-s + (−5.84 + 0.791i)15-s + (−6.64 − 3.83i)17-s + (−0.850 − 1.47i)19-s + (0.250 + 4.57i)21-s − 1.26i·23-s − 6.58·25-s + (2.04 + 4.77i)27-s + (−3.04 − 5.26i)29-s + (3.66 + 6.34i)31-s + ⋯
L(s)  = 1  + (−0.134 − 0.990i)3-s − 1.52i·5-s + (−0.996 − 0.0799i)7-s + (−0.963 + 0.266i)9-s + 1.66i·11-s + (0.545 + 0.314i)13-s + (−1.50 + 0.204i)15-s + (−1.61 − 0.931i)17-s + (−0.195 − 0.338i)19-s + (0.0546 + 0.998i)21-s − 0.264i·23-s − 1.31·25-s + (0.393 + 0.919i)27-s + (−0.564 − 0.978i)29-s + (0.657 + 1.13i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.471 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.471 - 0.882i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.471 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2375953907\)
\(L(\frac12)\) \(\approx\) \(0.2375953907\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.232 + 1.71i)T \)
7 \( 1 + (2.63 + 0.211i)T \)
good5 \( 1 + 3.40iT - 5T^{2} \)
11 \( 1 - 5.50iT - 11T^{2} \)
13 \( 1 + (-1.96 - 1.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.64 + 3.83i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.850 + 1.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.26iT - 23T^{2} \)
29 \( 1 + (3.04 + 5.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.66 - 6.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.928 + 1.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.99 - 2.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.68 - 3.85i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.822 - 1.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.90 - 5.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.48 + 6.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.64 - 5.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.13 - 4.12i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.47iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.96 + 1.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.67 + 9.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.67 + 2.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.903 - 0.521i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.144165672866984035110264362417, −8.773430647806394657348480123660, −7.63742923347836150031982456599, −6.84588671286080739304290606613, −6.16478145314348071941328911646, −4.92088696943347008728664211061, −4.32691120764782305247291634871, −2.61433336097242330259083922439, −1.52195635545302094409207684475, −0.10648227656325161106911851626, 2.60608730985015407598212651109, 3.42635248256182202784453999043, 3.96670390809455813210879985837, 5.66457361267712011402390227782, 6.20049764391988087949672537662, 6.84943793122915607743108813141, 8.279496915470700275019147902376, 8.903567425348861596278124904700, 9.895423749651336018515332223348, 10.63799024126374326040017614389

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