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Properties

Label 2-792-11.3-c1-0-4
Degree $2$
Conductor $792$
Sign $-0.162 - 0.986i$
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  + (1.68 + 1.22i)5-s + (−0.429 + 1.32i)7-s + (0.617 + 3.25i)11-s + (−4.68 + 3.40i)13-s + (−5.85 − 4.25i)17-s + (2.61 + 8.03i)19-s + 2.19·23-s + (−0.208 − 0.640i)25-s + (2.17 − 6.68i)29-s + (2.06 − 1.50i)31-s + (−2.33 + 1.70i)35-s + (−2.64 + 8.15i)37-s + (−0.331 − 1.01i)41-s − 7.59·43-s + (3.33 + 10.2i)47-s + ⋯
L(s)  = 1  + (0.752 + 0.546i)5-s + (−0.162 + 0.499i)7-s + (0.186 + 0.982i)11-s + (−1.29 + 0.944i)13-s + (−1.41 − 1.03i)17-s + (0.598 + 1.84i)19-s + 0.457·23-s + (−0.0416 − 0.128i)25-s + (0.403 − 1.24i)29-s + (0.371 − 0.269i)31-s + (−0.395 + 0.287i)35-s + (−0.435 + 1.34i)37-s + (−0.0517 − 0.159i)41-s − 1.15·43-s + (0.485 + 1.49i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891838 + 1.05092i\)
\(L(\frac12)\) \(\approx\) \(0.891838 + 1.05092i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (-0.617 - 3.25i)T \)
good5 \( 1 + (-1.68 - 1.22i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.429 - 1.32i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.68 - 3.40i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.85 + 4.25i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.61 - 8.03i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 + (-2.17 + 6.68i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.06 + 1.50i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.64 - 8.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.331 + 1.01i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.59T + 43T^{2} \)
47 \( 1 + (-3.33 - 10.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.445 + 0.323i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.24 + 9.97i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.56 - 6.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.71T + 67T^{2} \)
71 \( 1 + (-1.63 - 1.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.0934 - 0.287i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.97 - 2.89i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.62 - 6.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.07T + 89T^{2} \)
97 \( 1 + (-0.949 + 0.689i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.10926816326561391471513800025, −9.835002463724702861646273125459, −9.071044705741901062847629384781, −7.85322175132527537622497724741, −6.88724015237419915044653977287, −6.32496121018247900288848786848, −5.13464177611131598998770929854, −4.27590905227300107122852068392, −2.67554302189363647870801192938, −1.97288513836005409719664724762, 0.66292765484432402237097493486, 2.28124183832995027681187942496, 3.46102660538359257680428869049, 4.84922672645259991016271741471, 5.43337577976778525418911720521, 6.63485960234076261137367889178, 7.31435173878503625169637202384, 8.677105015773015835296803480443, 9.009111798376140560657979070651, 10.12270668760074241957702743640

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