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Properties

Label 2-1008-63.58-c1-0-18
Degree $2$
Conductor $1008$
Sign $-0.0579 - 0.998i$
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  + (0.914 + 1.47i)3-s + (0.891 + 1.54i)5-s + (2.54 − 0.727i)7-s + (−1.32 + 2.69i)9-s + (−2.80 + 4.86i)11-s + (3.14 − 5.43i)13-s + (−1.45 + 2.72i)15-s + (0.646 + 1.11i)17-s + (−0.559 + 0.968i)19-s + (3.39 + 3.07i)21-s + (3.80 + 6.59i)23-s + (0.909 − 1.57i)25-s + (−5.17 + 0.507i)27-s + (−1.57 − 2.72i)29-s − 1.00·31-s + ⋯
L(s)  = 1  + (0.527 + 0.849i)3-s + (0.398 + 0.690i)5-s + (0.961 − 0.274i)7-s + (−0.442 + 0.896i)9-s + (−0.846 + 1.46i)11-s + (0.870 − 1.50i)13-s + (−0.376 + 0.703i)15-s + (0.156 + 0.271i)17-s + (−0.128 + 0.222i)19-s + (0.741 + 0.671i)21-s + (0.794 + 1.37i)23-s + (0.181 − 0.315i)25-s + (−0.995 + 0.0977i)27-s + (−0.292 − 0.506i)29-s − 0.180·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0579 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0579 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0579 - 0.998i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.0579 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.171338339\)
\(L(\frac12)\) \(\approx\) \(2.171338339\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.914 - 1.47i)T \)
7 \( 1 + (-2.54 + 0.727i)T \)
good5 \( 1 + (-0.891 - 1.54i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.80 - 4.86i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.14 + 5.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.646 - 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.559 - 0.968i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.80 - 6.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.57 + 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.00T + 31T^{2} \)
37 \( 1 + (5.96 - 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.14 + 7.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.34 + 4.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.94T + 47T^{2} \)
53 \( 1 + (4.45 + 7.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.38T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 + (-5.67 - 9.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + (-1.60 - 2.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.404 - 0.700i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.10 - 1.91i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.32716820774759002999984135541, −9.569651688370796881311972104211, −8.374320645572892655432539769602, −7.86630504202505272210509535443, −7.00858821996802980900857481841, −5.54759972514720789591523553569, −5.04247192899903281636126557702, −3.88186281019019955832899351201, −2.91128135295261821338405287629, −1.81737253481718906078355526157, 1.00089668586942972933233971393, 2.03413363970366492925941399864, 3.18233085512007534366258464434, 4.52852519982993817552462506431, 5.53433291439425994314271591296, 6.31926718170863679769124792497, 7.33708358753466256210884818161, 8.310576850405077250493005611038, 8.806268348819999197446109281473, 9.260992102387933972037022584902

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