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Properties

Label 2-1008-21.2-c2-0-15
Degree $2$
Conductor $1008$
Sign $0.546 + 0.837i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.29 − 1.32i)5-s + (−5.75 + 3.98i)7-s + (1.93 − 1.11i)11-s + 2.75·13-s + (0.189 − 0.109i)17-s + (−6.60 + 11.4i)19-s + (17.4 + 10.0i)23-s + (−8.99 − 15.5i)25-s − 13.5i·29-s + (0.993 + 1.72i)31-s + (18.4 − 1.52i)35-s + (22.5 − 39.0i)37-s − 28.7i·41-s − 53.1·43-s + (62.8 + 36.2i)47-s + ⋯
L(s)  = 1  + (−0.458 − 0.264i)5-s + (−0.821 + 0.569i)7-s + (0.175 − 0.101i)11-s + 0.212·13-s + (0.0111 − 0.00642i)17-s + (−0.347 + 0.602i)19-s + (0.759 + 0.438i)23-s + (−0.359 − 0.623i)25-s − 0.468i·29-s + (0.0320 + 0.0555i)31-s + (0.527 − 0.0436i)35-s + (0.609 − 1.05i)37-s − 0.702i·41-s − 1.23·43-s + (1.33 + 0.771i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.546 + 0.837i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.226737107\)
\(L(\frac12)\) \(\approx\) \(1.226737107\)
\(L(2)\) 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 \( 1 + (5.75 - 3.98i)T \)
good5 \( 1 + (2.29 + 1.32i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.93 + 1.11i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 2.75T + 169T^{2} \)
17 \( 1 + (-0.189 + 0.109i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.60 - 11.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-17.4 - 10.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 13.5iT - 841T^{2} \)
31 \( 1 + (-0.993 - 1.72i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-22.5 + 39.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 28.7iT - 1.68e3T^{2} \)
43 \( 1 + 53.1T + 1.84e3T^{2} \)
47 \( 1 + (-62.8 - 36.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-52.0 + 30.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-56.8 + 32.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-37.7 + 65.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.1 + 27.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 17.2iT - 5.04e3T^{2} \)
73 \( 1 + (7.18 + 12.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-18.0 + 31.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 43.6iT - 6.88e3T^{2} \)
89 \( 1 + (10.2 + 5.89i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−9.569343069145330586059661166496, −8.822043758729282220107529376332, −8.081608324919568094914314351705, −7.08828626659582677669582077162, −6.18710006134606412983931114002, −5.41662320469129774468777953264, −4.19122756682335983311018420939, −3.36171915859371896615824332509, −2.14393694797664431222981357013, −0.48713303129629189199347684165, 0.963142798183163988793493155507, 2.66902674079846445388178420454, 3.61719588151897747847955673767, 4.47527369568378502149683539191, 5.66297680286050715863611924786, 6.78354524367202253134632125674, 7.13835027537563116968917933523, 8.271214115182228095653607970454, 9.086598741037078204311007212678, 9.965105742928970836922788609687

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