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Properties

Label 2-1008-63.25-c1-0-26
Degree $2$
Conductor $1008$
Sign $0.638 + 0.769i$
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  + (−0.933 + 1.45i)3-s + (−0.296 + 0.514i)5-s + (−2.32 + 1.26i)7-s + (−1.25 − 2.72i)9-s + (−0.296 − 0.514i)11-s + (−1.25 − 2.17i)13-s + (−0.472 − 0.912i)15-s + (1.46 − 2.52i)17-s + (−2.69 − 4.66i)19-s + (0.323 − 4.57i)21-s + (2.23 − 3.86i)23-s + (2.32 + 4.02i)25-s + (5.14 + 0.708i)27-s + (−3.09 + 5.36i)29-s + 7.86·31-s + ⋯
L(s)  = 1  + (−0.538 + 0.842i)3-s + (−0.132 + 0.229i)5-s + (−0.878 + 0.478i)7-s + (−0.419 − 0.907i)9-s + (−0.0894 − 0.154i)11-s + (−0.348 − 0.603i)13-s + (−0.122 − 0.235i)15-s + (0.354 − 0.613i)17-s + (−0.617 − 1.06i)19-s + (0.0706 − 0.997i)21-s + (0.465 − 0.805i)23-s + (0.464 + 0.804i)25-s + (0.990 + 0.136i)27-s + (−0.575 + 0.996i)29-s + 1.41·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.638 + 0.769i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6988914604\)
\(L(\frac12)\) \(\approx\) \(0.6988914604\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.933 - 1.45i)T \)
7 \( 1 + (2.32 - 1.26i)T \)
good5 \( 1 + (0.296 - 0.514i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.296 + 0.514i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.25 + 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.46 + 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.69 + 4.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.23 + 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.09 - 5.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.86T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.58 + 9.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + (-4.02 + 6.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.64T + 59T^{2} \)
61 \( 1 + 6.64T + 61T^{2} \)
67 \( 1 - 1.91T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (-3.95 + 6.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 + (3.85 - 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.21 + 10.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.86 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.851419965500641769618782783101, −9.191735184773029886404185040496, −8.432536494702805238050536230713, −7.06728908982733059560864662888, −6.43897247883904184335765597816, −5.39702463163573144640464018722, −4.74233028439509593182627061415, −3.42208860322518367128159568356, −2.74949312975099090724305925870, −0.38101654141231479877230176497, 1.19369179840054501224239449699, 2.54119648073484647939638406202, 3.89020791889088343854179947962, 4.89397526299315625053911525482, 6.17320075060794553583574608279, 6.46895300906658420863929977530, 7.63379415787200545327559284115, 8.115877464592453149040165543548, 9.355759395206088011562605182671, 10.12147673975391372888831041843

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