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Properties

Label 2-1008-252.103-c1-0-42
Degree $2$
Conductor $1008$
Sign $-0.583 + 0.812i$
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  + (1.60 − 0.649i)3-s + (−2.10 + 1.21i)5-s + (−0.347 − 2.62i)7-s + (2.15 − 2.08i)9-s + (−3.88 − 2.24i)11-s + (0.395 + 0.228i)13-s + (−2.59 + 3.32i)15-s + (−1.45 + 0.839i)17-s + (3.17 − 5.50i)19-s + (−2.26 − 3.98i)21-s + (−3.03 + 1.75i)23-s + (0.456 − 0.791i)25-s + (2.10 − 4.75i)27-s + (−1.38 − 2.39i)29-s − 8.92·31-s + ⋯
L(s)  = 1  + (0.926 − 0.375i)3-s + (−0.941 + 0.543i)5-s + (−0.131 − 0.991i)7-s + (0.718 − 0.695i)9-s + (−1.17 − 0.676i)11-s + (0.109 + 0.0633i)13-s + (−0.669 + 0.857i)15-s + (−0.352 + 0.203i)17-s + (0.728 − 1.26i)19-s + (−0.493 − 0.869i)21-s + (−0.633 + 0.365i)23-s + (0.0913 − 0.158i)25-s + (0.405 − 0.914i)27-s + (−0.256 − 0.444i)29-s − 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.583 + 0.812i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.583 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.178638155\)
\(L(\frac12)\) \(\approx\) \(1.178638155\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 + 0.649i)T \)
7 \( 1 + (0.347 + 2.62i)T \)
good5 \( 1 + (2.10 - 1.21i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.88 + 2.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.395 - 0.228i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.45 - 0.839i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.17 + 5.50i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.03 - 1.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.38 + 2.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.92T + 31T^{2} \)
37 \( 1 + (-0.463 + 0.802i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.08 + 5.24i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.87 + 5.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.39T + 47T^{2} \)
53 \( 1 + (4.91 + 8.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.45T + 59T^{2} \)
61 \( 1 - 5.31iT - 61T^{2} \)
67 \( 1 + 13.6iT - 67T^{2} \)
71 \( 1 - 1.59iT - 71T^{2} \)
73 \( 1 + (-9.31 + 5.38i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (-0.657 - 1.13i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-13.5 - 7.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.69 - 3.28i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.590445442007636070423821107869, −8.726618597800248468858445907523, −7.73063150998388291123441408804, −7.46058314134945475819201299869, −6.64351375023008838598730626391, −5.26391132921838834422422297424, −3.89156490084221060328145286948, −3.44392482463568262913734425086, −2.28102585330534946568732486031, −0.45768530327829969827267167315, 1.92929023939651085734522329940, 2.99573254374262795278221128393, 3.99906017115632679720631750043, 4.90433492029804972868868072917, 5.75779792973045366491734615709, 7.30122498947783341160759486704, 7.919058315256898436540680535177, 8.518511523005002428338413783071, 9.340878499530071327392540249931, 10.06435629663764629457081077396

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