Properties

Degree $2$
Conductor $1008$
Sign $0.0633 - 0.997i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3 + 1.73i)5-s + (0.5 − 2.59i)7-s + (3 + 1.73i)11-s − 1.73i·13-s + (2.5 + 4.33i)19-s + (−6 + 3.46i)23-s + (3.5 − 6.06i)25-s + (−2.5 + 4.33i)31-s + (3 + 8.66i)35-s + (5.5 + 9.52i)37-s + 3.46i·41-s + 8.66i·43-s + (3 + 5.19i)47-s + (−6.5 − 2.59i)49-s + (6 − 10.3i)53-s + ⋯
L(s)  = 1  + (−1.34 + 0.774i)5-s + (0.188 − 0.981i)7-s + (0.904 + 0.522i)11-s − 0.480i·13-s + (0.573 + 0.993i)19-s + (−1.25 + 0.722i)23-s + (0.700 − 1.21i)25-s + (−0.449 + 0.777i)31-s + (0.507 + 1.46i)35-s + (0.904 + 1.56i)37-s + 0.541i·41-s + 1.32i·43-s + (0.437 + 0.757i)47-s + (−0.928 − 0.371i)49-s + (0.824 − 1.42i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.0633 - 0.997i$
Motivic weight: \(1\)
Character: $\chi_{1008} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.012069709\)
\(L(\frac12)\) \(\approx\) \(1.012069709\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6 - 3.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12 - 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.5 + 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−10.24202483320007231712126742819, −9.513056003111193065572896681447, −8.097822301255009424413827706456, −7.73280869412373413483983015680, −6.97363138599567634218341814174, −6.09252956179449645909593151779, −4.61743079853058161984363137300, −3.87185642525035786147726792906, −3.17003521237528029480590233449, −1.32996170163245148892856759690, 0.51546316417718536849644538196, 2.19679293088573054801475842264, 3.66220981038429346387326354516, 4.34882733225343345910354384387, 5.36500534983608535166264534980, 6.32234341339727357080220407994, 7.43319898673919815538413594361, 8.179619022526110775691749660739, 8.997927582986173972982356615092, 9.328350962423188572928703700507

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