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Properties

Label 2-1008-252.103-c1-0-9
Degree $2$
Conductor $1008$
Sign $0.463 - 0.886i$
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  + (−0.908 − 1.47i)3-s + (−0.243 + 0.140i)5-s + (2.20 + 1.46i)7-s + (−1.35 + 2.67i)9-s + (−2.44 − 1.41i)11-s + (4.06 + 2.34i)13-s + (0.427 + 0.231i)15-s + (−7.00 + 4.04i)17-s + (−0.474 + 0.821i)19-s + (0.155 − 4.57i)21-s + (−0.339 + 0.196i)23-s + (−2.46 + 4.26i)25-s + (5.17 − 0.440i)27-s + (1.51 + 2.61i)29-s − 2.12·31-s + ⋯
L(s)  = 1  + (−0.524 − 0.851i)3-s + (−0.108 + 0.0627i)5-s + (0.833 + 0.552i)7-s + (−0.450 + 0.892i)9-s + (−0.736 − 0.425i)11-s + (1.12 + 0.651i)13-s + (0.110 + 0.0596i)15-s + (−1.69 + 0.980i)17-s + (−0.108 + 0.188i)19-s + (0.0339 − 0.999i)21-s + (−0.0708 + 0.0409i)23-s + (−0.492 + 0.852i)25-s + (0.996 − 0.0848i)27-s + (0.280 + 0.486i)29-s − 0.382·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.463 - 0.886i$
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: \((2,\ 1008,\ (\ :1/2),\ 0.463 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9968065223\)
\(L(\frac12)\) \(\approx\) \(0.9968065223\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.908 + 1.47i)T \)
7 \( 1 + (-2.20 - 1.46i)T \)
good5 \( 1 + (0.243 - 0.140i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.44 + 1.41i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.06 - 2.34i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (7.00 - 4.04i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.474 - 0.821i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.339 - 0.196i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.51 - 2.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + (2.43 - 4.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.478 - 0.276i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.28 + 2.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.78T + 47T^{2} \)
53 \( 1 + (-6.21 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 11.1iT - 61T^{2} \)
67 \( 1 - 9.07iT - 67T^{2} \)
71 \( 1 + 1.54iT - 71T^{2} \)
73 \( 1 + (0.542 - 0.313i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.4iT - 79T^{2} \)
83 \( 1 + (5.30 + 9.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.90 - 5.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.6 + 7.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.45651075655749455071164901566, −8.856182073663570999557583710231, −8.506265522930620608570432452896, −7.62373013377256291100575364464, −6.63256467562417278982645353627, −5.91181605796903799053032629147, −5.08792087540350644857009299829, −3.95251996718946983004473428568, −2.39102586952971225155448460396, −1.47564376949384279737285671371, 0.49653027218673668850024879947, 2.36543378237490545638935319254, 3.80627930178872928789782611169, 4.56369759274819911032471818855, 5.29604146336281514548945661130, 6.31411333349881345730716142498, 7.29487494507704639882094714964, 8.311491084531845539948233477152, 8.966617629891859050033408690374, 10.04787858325900886324593571487

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