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Properties

Label 2-1008-252.103-c1-0-15
Degree $2$
Conductor $1008$
Sign $0.946 + 0.323i$
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  + (−1.02 + 1.39i)3-s + (−3.67 + 2.12i)5-s + (−2.04 − 1.67i)7-s + (−0.911 − 2.85i)9-s + (1.49 + 0.863i)11-s + (−2.74 − 1.58i)13-s + (0.788 − 7.30i)15-s + (3.64 − 2.10i)17-s + (−3.66 + 6.35i)19-s + (4.43 − 1.15i)21-s + (−3.84 + 2.21i)23-s + (6.50 − 11.2i)25-s + (4.92 + 1.64i)27-s + (−0.0835 − 0.144i)29-s − 1.94·31-s + ⋯
L(s)  = 1  + (−0.590 + 0.807i)3-s + (−1.64 + 0.948i)5-s + (−0.774 − 0.633i)7-s + (−0.303 − 0.952i)9-s + (0.451 + 0.260i)11-s + (−0.760 − 0.439i)13-s + (0.203 − 1.88i)15-s + (0.883 − 0.509i)17-s + (−0.841 + 1.45i)19-s + (0.967 − 0.251i)21-s + (−0.801 + 0.462i)23-s + (1.30 − 2.25i)25-s + (0.948 + 0.316i)27-s + (−0.0155 − 0.0268i)29-s − 0.349·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.946 + 0.323i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.946 + 0.323i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4582361285\)
\(L(\frac12)\) \(\approx\) \(0.4582361285\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 1.39i)T \)
7 \( 1 + (2.04 + 1.67i)T \)
good5 \( 1 + (3.67 - 2.12i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.49 - 0.863i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.74 + 1.58i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.64 + 2.10i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.66 - 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.84 - 2.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0835 + 0.144i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.94T + 31T^{2} \)
37 \( 1 + (-5.04 + 8.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.21 - 3.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.51 - 2.03i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + (-2.59 - 4.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.67T + 59T^{2} \)
61 \( 1 + 7.00iT - 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 - 2.30iT - 71T^{2} \)
73 \( 1 + (-2.81 + 1.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.38iT - 79T^{2} \)
83 \( 1 + (0.341 + 0.591i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.57 + 4.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.8 - 6.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.11252850448361415051518891481, −9.385761172429663929412991149476, −7.988666387741094499130213390432, −7.44390045410187137242269985398, −6.57900471550314536821373854081, −5.66909792600172173111042945728, −4.19750208295560211316994765814, −3.88094116163272454966980843952, −2.95999026833973498679647088171, −0.33830011012120310935790974533, 0.812856715830826289297012798583, 2.52108139017529561081308845694, 3.89353100092875136576486583351, 4.76188073377022012830072113238, 5.74111788706336728464161283189, 6.76333986603952033184395678371, 7.43015768579673714282493817188, 8.407790429628111116803828157642, 8.830293150296047905644320661536, 10.00511912294084355862413888087

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