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Properties

Label 2-792-88.43-c1-0-53
Degree $2$
Conductor $792$
Sign $-0.0708 + 0.997i$
Analytic cond. $6.32415$
Root an. cond. $2.51478$
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  + (1.37 − 0.331i)2-s + (1.78 − 0.910i)4-s − 2i·5-s − 3.02·7-s + (2.14 − 1.84i)8-s + (−0.662 − 2.74i)10-s + (2.33 − 2.35i)11-s − 1.32·13-s + (−4.15 + i)14-s + (2.34 − 3.24i)16-s + 1.69i·17-s − 7.04i·19-s + (−1.82 − 3.56i)20-s + (2.42 − 4.01i)22-s + 3.12i·23-s + ⋯
L(s)  = 1  + (0.972 − 0.234i)2-s + (0.890 − 0.455i)4-s − 0.894i·5-s − 1.14·7-s + (0.759 − 0.650i)8-s + (−0.209 − 0.869i)10-s + (0.703 − 0.711i)11-s − 0.367·13-s + (−1.10 + 0.267i)14-s + (0.585 − 0.810i)16-s + 0.411i·17-s − 1.61i·19-s + (−0.407 − 0.796i)20-s + (0.517 − 0.855i)22-s + 0.651i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0708 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0708 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.0708 + 0.997i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.0708 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69903 - 1.82398i\)
\(L(\frac12)\) \(\approx\) \(1.69903 - 1.82398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.331i)T \)
3 \( 1 \)
11 \( 1 + (-2.33 + 2.35i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 3.02T + 7T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 - 1.69iT - 17T^{2} \)
19 \( 1 + 7.04iT - 19T^{2} \)
23 \( 1 - 3.12iT - 23T^{2} \)
29 \( 1 + 1.54T + 29T^{2} \)
31 \( 1 + 8.30iT - 31T^{2} \)
37 \( 1 - 4.66iT - 37T^{2} \)
41 \( 1 - 7.73iT - 41T^{2} \)
43 \( 1 - 3.95iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 8.24iT - 53T^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 8.10T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 - 12.2iT - 71T^{2} \)
73 \( 1 - 3.08iT - 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 - 4.71iT - 83T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−9.978225098031570442100680291378, −9.405371030402195328498584238690, −8.444036836148266317462680679572, −7.14285902804149187854687637685, −6.38933160602335450616910029308, −5.54771898491111071843290584927, −4.56350401737221087426400089641, −3.63201883964075623497015072230, −2.60058402241511192245488348667, −0.929145860526413366207726639452, 2.11432535170431634640980757008, 3.25164366940671823449869811904, 3.92877573261394171543406793076, 5.20785551174675476939616160079, 6.26626477464054031086394477672, 6.84868152861629510801082287232, 7.47049250883496927770914448973, 8.765376941351285815712657042725, 9.957951140350925519897057147447, 10.45096341739566612824601300968

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