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Properties

Label 2-1008-63.4-c1-0-31
Degree $2$
Conductor $1008$
Sign $-0.764 + 0.644i$
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  + (−1.73 + 0.0789i)3-s − 0.460·5-s + (−2.25 − 1.38i)7-s + (2.98 − 0.273i)9-s + 3.64·11-s + (0.730 − 1.26i)13-s + (0.796 − 0.0363i)15-s + (−1.86 + 3.23i)17-s + (2.02 + 3.51i)19-s + (4.01 + 2.20i)21-s − 1.13·23-s − 4.78·25-s + (−5.14 + 0.708i)27-s + (−4.48 − 7.77i)29-s + (−0.257 − 0.445i)31-s + ⋯
L(s)  = 1  + (−0.998 + 0.0455i)3-s − 0.205·5-s + (−0.853 − 0.521i)7-s + (0.995 − 0.0910i)9-s + 1.09·11-s + (0.202 − 0.350i)13-s + (0.205 − 0.00938i)15-s + (−0.452 + 0.784i)17-s + (0.465 + 0.805i)19-s + (0.876 + 0.482i)21-s − 0.236·23-s − 0.957·25-s + (−0.990 + 0.136i)27-s + (−0.833 − 1.44i)29-s + (−0.0462 − 0.0800i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4006352662\)
\(L(\frac12)\) \(\approx\) \(0.4006352662\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0789i)T \)
7 \( 1 + (2.25 + 1.38i)T \)
good5 \( 1 + 0.460T + 5T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (-0.730 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 3.51i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.13T + 23T^{2} \)
29 \( 1 + (4.48 + 7.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.257 + 0.445i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.55 + 7.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.472 - 0.819i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.16 + 2.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.21 + 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.44 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.04 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.16 + 2.00i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 + (6.62 - 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.50 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.32 + 5.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.36 + 2.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 + 9.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.905694074114242719893325037915, −8.969586908571796919051900080132, −7.77962536111066591537894230726, −6.96483344040998626347428211401, −6.16541151867950841204177029123, −5.55950860449618476541696515471, −4.01518092750174708105673252263, −3.77606647391854451661155897704, −1.73673665905131937270034253148, −0.21725722712362141910799107253, 1.46888033376599572451699296229, 3.10029542898103028857316454742, 4.20748437566498736808595903149, 5.15905943986277968099351958805, 6.15004319458635247703476331284, 6.74565765151101268822082364419, 7.49591318772300041347099260095, 8.984125117556910209550034391000, 9.359941409415027155559225842119, 10.31459945233086739782356200622

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