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Properties

Label 2-1008-252.103-c1-0-46
Degree $2$
Conductor $1008$
Sign $-0.818 - 0.574i$
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  − 1.73i·3-s + (−0.621 + 0.358i)5-s + (−1 − 2.44i)7-s − 2.99·9-s + (1.5 + 0.866i)11-s + (−3.62 − 2.09i)13-s + (0.621 + 1.07i)15-s + (−2.74 + 1.58i)17-s + (−0.5 + 0.866i)19-s + (−4.24 + 1.73i)21-s + (2.37 − 1.37i)23-s + (−2.24 + 3.88i)25-s + 5.19i·27-s + (0.621 + 1.07i)29-s − 4·31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−0.277 + 0.160i)5-s + (−0.377 − 0.925i)7-s − 0.999·9-s + (0.452 + 0.261i)11-s + (−1.00 − 0.579i)13-s + (0.160 + 0.277i)15-s + (−0.665 + 0.384i)17-s + (−0.114 + 0.198i)19-s + (−0.925 + 0.377i)21-s + (0.495 − 0.286i)23-s + (−0.448 + 0.776i)25-s + 0.999i·27-s + (0.115 + 0.199i)29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.818 - 0.574i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3069191353\)
\(L(\frac12)\) \(\approx\) \(0.3069191353\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT \)
7 \( 1 + (1 + 2.44i)T \)
good5 \( 1 + (0.621 - 0.358i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.62 + 2.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.74 - 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.37 + 1.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.621 - 1.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.62 + 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.74 + 5.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.74 - 3.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + (0.621 + 1.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.02iT - 79T^{2} \)
83 \( 1 + (-5.74 - 9.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (14.2 + 8.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.74 - 3.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.423985735913397837799284685943, −8.516780930678185612986936226648, −7.48685893647279673662368995884, −7.13956181482599080980308426806, −6.28635967678504969859230528914, −5.18857338717967522545908782128, −3.98206653888076263213656094392, −2.94629856418937598033931406224, −1.64858666262960680356715500622, −0.13275566962847575912898837061, 2.28570804520946201145481766756, 3.29451855543711222353244721258, 4.40100400139387550154509181759, 5.11267688626246382646835678190, 6.10133393675261646504632070141, 7.00400775233247991759206791301, 8.275372210662248583009331383041, 8.971635386883574483563747552097, 9.537922618486151046668524012020, 10.28230202430087247212523190467

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