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Properties

Label 2-1008-252.103-c1-0-23
Degree $2$
Conductor $1008$
Sign $-0.381 + 0.924i$
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  + (−1.29 − 1.15i)3-s + (−2.47 + 1.43i)5-s + (−1.38 + 2.25i)7-s + (0.340 + 2.98i)9-s + (3.57 + 2.06i)11-s + (−3.14 − 1.81i)13-s + (4.85 + 1.00i)15-s + (−3.36 + 1.94i)17-s + (3.57 − 6.18i)19-s + (4.39 − 1.30i)21-s + (5.18 − 2.99i)23-s + (1.60 − 2.77i)25-s + (2.99 − 4.24i)27-s + (−2.87 − 4.98i)29-s − 10.4·31-s + ⋯
L(s)  = 1  + (−0.746 − 0.665i)3-s + (−1.10 + 0.640i)5-s + (−0.524 + 0.851i)7-s + (0.113 + 0.993i)9-s + (1.07 + 0.622i)11-s + (−0.870 − 0.502i)13-s + (1.25 + 0.260i)15-s + (−0.815 + 0.470i)17-s + (0.819 − 1.41i)19-s + (0.958 − 0.285i)21-s + (1.08 − 0.623i)23-s + (0.320 − 0.554i)25-s + (0.576 − 0.816i)27-s + (−0.534 − 0.925i)29-s − 1.88·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.381 + 0.924i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3771696297\)
\(L(\frac12)\) \(\approx\) \(0.3771696297\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 1.15i)T \)
7 \( 1 + (1.38 - 2.25i)T \)
good5 \( 1 + (2.47 - 1.43i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.57 - 2.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.14 + 1.81i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.36 - 1.94i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.57 + 6.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.18 + 2.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.87 + 4.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (2.02 - 3.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.64 - 1.52i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.533 - 0.308i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.07T + 47T^{2} \)
53 \( 1 + (2.65 + 4.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.29T + 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + 1.58iT - 67T^{2} \)
71 \( 1 + 5.90iT - 71T^{2} \)
73 \( 1 + (6.78 - 3.91i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.05iT - 79T^{2} \)
83 \( 1 + (1.69 + 2.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.80 - 5.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.52 + 1.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.658884504175659732335670153211, −8.899348007064104172809606248734, −7.76463023328740519598448731582, −7.00242725302921005079163249325, −6.62024315308823756501001159654, −5.41535527541157584467959142944, −4.50786839100090389469082158813, −3.24451596359824459074436198066, −2.14050402999342593830449212308, −0.22634838315551324725614623930, 1.09835238700767701680929678964, 3.55332653250596707429245423332, 3.91051029879628082575495071222, 4.88566755414890204142440588694, 5.81918467127957854423156483250, 7.04452672733819135522913068249, 7.42182246864540251885322146183, 8.948130880969475133261651283130, 9.247802753475454392243337956690, 10.27774031524155733641486504259

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