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Properties

Label 2-1008-63.59-c1-0-25
Degree $2$
Conductor $1008$
Sign $0.802 - 0.596i$
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  + (1.11 + 1.32i)3-s + 3.53·5-s + (−1.30 − 2.30i)7-s + (−0.525 + 2.95i)9-s − 1.81i·11-s + (2.78 + 1.60i)13-s + (3.93 + 4.69i)15-s + (−2.93 + 5.08i)17-s + (2.09 − 1.20i)19-s + (1.61 − 4.28i)21-s + 3.84i·23-s + 7.48·25-s + (−4.50 + 2.58i)27-s + (5.79 − 3.34i)29-s + (4.21 − 2.43i)31-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)3-s + 1.58·5-s + (−0.491 − 0.870i)7-s + (−0.175 + 0.984i)9-s − 0.548i·11-s + (0.771 + 0.445i)13-s + (1.01 + 1.21i)15-s + (−0.711 + 1.23i)17-s + (0.479 − 0.277i)19-s + (0.351 − 0.936i)21-s + 0.802i·23-s + 1.49·25-s + (−0.867 + 0.497i)27-s + (1.07 − 0.620i)29-s + (0.757 − 0.437i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.802 - 0.596i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.540495829\)
\(L(\frac12)\) \(\approx\) \(2.540495829\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (1.30 + 2.30i)T \)
good5 \( 1 - 3.53T + 5T^{2} \)
11 \( 1 + 1.81iT - 11T^{2} \)
13 \( 1 + (-2.78 - 1.60i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.93 - 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.09 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.84iT - 23T^{2} \)
29 \( 1 + (-5.79 + 3.34i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.21 + 2.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.905 + 1.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.03 + 8.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.36 - 4.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.96 - 6.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.26 - 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.40 + 7.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.98 + 5.18i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.92 + 6.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.62iT - 71T^{2} \)
73 \( 1 + (11.7 + 6.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.921 - 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.61 - 9.72i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.89 + 11.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.7 - 7.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.955740769703862966989782043043, −9.335975293512208177814105094540, −8.676351540455049080116677458638, −7.66290155096460377655683350170, −6.37145948077779619772616554009, −5.93049679828190202342903341750, −4.67349872288630225699432846976, −3.76283933144951515686854505625, −2.72236137150249945639958777889, −1.51448120926508876158409098759, 1.30486726928315987991841108504, 2.47845664390968039462273998826, 3.00000847370596078431687325890, 4.78825981571653142184992097978, 5.86337657955690766694244344135, 6.41867898668314672027018135333, 7.20086948327647742917211007195, 8.509500706260498795532955650581, 8.982724877216093983775592574365, 9.739207314110874462318959355827

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