Properties

Label 2-1008-63.58-c1-0-34
Degree $2$
Conductor $1008$
Sign $0.381 + 0.924i$
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  + (1.64 − 0.545i)3-s + (−0.794 − 1.37i)5-s + (−1.23 + 2.33i)7-s + (2.40 − 1.79i)9-s + (−0.794 + 1.37i)11-s + (2.40 − 4.16i)13-s + (−2.05 − 1.82i)15-s + (−2.69 − 4.67i)17-s + (3.54 − 6.14i)19-s + (−0.761 + 4.51i)21-s + (0.150 + 0.260i)23-s + (1.23 − 2.14i)25-s + (2.97 − 4.25i)27-s + (4.13 + 7.16i)29-s + 2.71·31-s + ⋯
L(s)  = 1  + (0.949 − 0.314i)3-s + (−0.355 − 0.615i)5-s + (−0.468 + 0.883i)7-s + (0.801 − 0.597i)9-s + (−0.239 + 0.414i)11-s + (0.667 − 1.15i)13-s + (−0.530 − 0.472i)15-s + (−0.654 − 1.13i)17-s + (0.814 − 1.41i)19-s + (−0.166 + 0.986i)21-s + (0.0313 + 0.0542i)23-s + (0.247 − 0.429i)25-s + (0.572 − 0.819i)27-s + (0.768 + 1.33i)29-s + 0.487·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.381 + 0.924i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.971908666\)
\(L(\frac12)\) \(\approx\) \(1.971908666\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.545i)T \)
7 \( 1 + (1.23 - 2.33i)T \)
good5 \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.794 - 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.54 + 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.150 - 0.260i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.71T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.833 - 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.66T + 47T^{2} \)
53 \( 1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.47T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.38T + 79T^{2} \)
83 \( 1 + (1.18 + 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.476864462198204342438253484012, −8.921951959289994230845196693530, −8.333884178066667171072430572951, −7.37955748933619450663014303786, −6.61837400366382200691221802254, −5.35813398189847867956567741615, −4.53735514978530019890637063504, −3.14099252249708479365203334680, −2.57490836710793846027632685661, −0.867995735331923963404615627086, 1.58519145803084804175052005472, 3.02028627755577217815547505454, 3.80587243408054002686252434227, 4.42408101901510414349463888895, 6.08797020147025269176192099419, 6.81782502504492113906862851829, 7.78182592752254096082298172937, 8.339999466315244440137123589916, 9.363145636115573424240018203325, 10.09964724662244201822311452519

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