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Properties

Label 2-1008-48.11-c1-0-34
Degree $2$
Conductor $1008$
Sign $0.900 - 0.435i$
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  + (0.614 + 1.27i)2-s + (−1.24 + 1.56i)4-s + (2.62 − 2.62i)5-s + 7-s + (−2.75 − 0.623i)8-s + (4.96 + 1.73i)10-s + (0.583 + 0.583i)11-s + (1.76 − 1.76i)13-s + (0.614 + 1.27i)14-s + (−0.901 − 3.89i)16-s − 3.63i·17-s + (0.963 + 0.963i)19-s + (0.842 + 7.38i)20-s + (−0.384 + 1.10i)22-s − 3.77i·23-s + ⋯
L(s)  = 1  + (0.434 + 0.900i)2-s + (−0.622 + 0.782i)4-s + (1.17 − 1.17i)5-s + 0.377·7-s + (−0.975 − 0.220i)8-s + (1.56 + 0.547i)10-s + (0.175 + 0.175i)11-s + (0.489 − 0.489i)13-s + (0.164 + 0.340i)14-s + (−0.225 − 0.974i)16-s − 0.882i·17-s + (0.221 + 0.221i)19-s + (0.188 + 1.65i)20-s + (−0.0819 + 0.234i)22-s − 0.786i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.395181612\)
\(L(\frac12)\) \(\approx\) \(2.395181612\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.614 - 1.27i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (-2.62 + 2.62i)T - 5iT^{2} \)
11 \( 1 + (-0.583 - 0.583i)T + 11iT^{2} \)
13 \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \)
17 \( 1 + 3.63iT - 17T^{2} \)
19 \( 1 + (-0.963 - 0.963i)T + 19iT^{2} \)
23 \( 1 + 3.77iT - 23T^{2} \)
29 \( 1 + (-4.76 - 4.76i)T + 29iT^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 + (4.66 + 4.66i)T + 37iT^{2} \)
41 \( 1 - 3.83T + 41T^{2} \)
43 \( 1 + (3.45 - 3.45i)T - 43iT^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + (-5.48 + 5.48i)T - 53iT^{2} \)
59 \( 1 + (-4.40 - 4.40i)T + 59iT^{2} \)
61 \( 1 + (7.99 - 7.99i)T - 61iT^{2} \)
67 \( 1 + (11.3 + 11.3i)T + 67iT^{2} \)
71 \( 1 - 5.85iT - 71T^{2} \)
73 \( 1 + 0.564iT - 73T^{2} \)
79 \( 1 - 16.4iT - 79T^{2} \)
83 \( 1 + (6.92 - 6.92i)T - 83iT^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 8.64T + 97T^{2} \)
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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

−9.767708090056881279403376251338, −8.850118943647348196438819668718, −8.583061166201953001429171487213, −7.42758084375156196536467982843, −6.49932852250584570644989594649, −5.54135351259300265449202710904, −5.09127549426207641540093234713, −4.16456243717927711061445900775, −2.70281358869433963002778537300, −1.11516269291750066530316331908, 1.51270981014568608348665209913, 2.41629694085311040433892866450, 3.41764231071290358087855383498, 4.45428958238628626663686965388, 5.77315821232466273585114887263, 6.12385951255965617202815632019, 7.19758631236992732720232543325, 8.536453918499068219157069605472, 9.371288362448195426100409626654, 10.15491028868440648347595110270

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