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Properties

Label 2-1008-252.103-c1-0-24
Degree $2$
Conductor $1008$
Sign $0.759 - 0.650i$
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  + (−1.00 + 1.41i)3-s + (1.92 − 1.11i)5-s + (1.98 + 1.74i)7-s + (−0.981 − 2.83i)9-s + (3.11 + 1.79i)11-s + (−0.699 − 0.404i)13-s + (−0.365 + 3.82i)15-s + (1.37 − 0.796i)17-s + (1.63 − 2.83i)19-s + (−4.46 + 1.05i)21-s + (1.46 − 0.844i)23-s + (−0.0343 + 0.0595i)25-s + (4.98 + 1.46i)27-s + (−2.13 − 3.69i)29-s + 1.26·31-s + ⋯
L(s)  = 1  + (−0.579 + 0.814i)3-s + (0.860 − 0.496i)5-s + (0.751 + 0.659i)7-s + (−0.327 − 0.944i)9-s + (0.939 + 0.542i)11-s + (−0.194 − 0.112i)13-s + (−0.0942 + 0.988i)15-s + (0.334 − 0.193i)17-s + (0.375 − 0.649i)19-s + (−0.973 + 0.229i)21-s + (0.305 − 0.176i)23-s + (−0.00687 + 0.0119i)25-s + (0.959 + 0.281i)27-s + (−0.395 − 0.685i)29-s + 0.226·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.759 - 0.650i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.759 - 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.761327138\)
\(L(\frac12)\) \(\approx\) \(1.761327138\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.00 - 1.41i)T \)
7 \( 1 + (-1.98 - 1.74i)T \)
good5 \( 1 + (-1.92 + 1.11i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.11 - 1.79i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.699 + 0.404i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.37 + 0.796i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.63 + 2.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.46 + 0.844i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.13 + 3.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + (-3.94 + 6.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.57 - 4.37i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.47 - 4.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.665T + 47T^{2} \)
53 \( 1 + (-6.27 - 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.30T + 59T^{2} \)
61 \( 1 + 8.59iT - 61T^{2} \)
67 \( 1 - 0.102iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (5.30 - 3.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.31iT - 79T^{2} \)
83 \( 1 + (-5.33 - 9.23i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.69 - 0.981i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-15.3 + 8.84i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.772610479941193043697093799070, −9.426500409629572379849729739190, −8.746960236423055372947461548258, −7.55656461118735327805373845073, −6.36912619266270165345018844949, −5.61444637677482607118944957851, −4.92083023909959395488929553519, −4.11459817238410290061482525488, −2.60921554355727616294124962784, −1.24462808144468192119226480131, 1.11942063539002821846196458220, 2.04402473279739957732875137089, 3.48966606498906026558775691241, 4.81235903478217552366676325972, 5.76766477418993049954439852534, 6.44999392971863024165497004021, 7.24576982143082518207115255368, 8.026618847345753685526671851874, 9.003438351229241467423734997077, 10.13615516637901075570095855181

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