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Properties

Label 2-792-24.11-c1-0-4
Degree $2$
Conductor $792$
Sign $-0.995 - 0.0916i$
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  + (0.401 + 1.35i)2-s + (−1.67 + 1.08i)4-s + 0.138·5-s − 0.456i·7-s + (−2.15 − 1.83i)8-s + (0.0557 + 0.188i)10-s + i·11-s + 5.11i·13-s + (0.619 − 0.183i)14-s + (1.62 − 3.65i)16-s + 6.17i·17-s + 3.15·19-s + (−0.232 + 0.151i)20-s + (−1.35 + 0.401i)22-s − 9.41·23-s + ⋯
L(s)  = 1  + (0.283 + 0.958i)2-s + (−0.838 + 0.544i)4-s + 0.0620·5-s − 0.172i·7-s + (−0.760 − 0.649i)8-s + (0.0176 + 0.0595i)10-s + 0.301i·11-s + 1.41i·13-s + (0.165 − 0.0490i)14-s + (0.407 − 0.913i)16-s + 1.49i·17-s + 0.722·19-s + (−0.0520 + 0.0338i)20-s + (−0.289 + 0.0855i)22-s − 1.96·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.995 - 0.0916i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.995 - 0.0916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0512727 + 1.11681i\)
\(L(\frac12)\) \(\approx\) \(0.0512727 + 1.11681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.401 - 1.35i)T \)
3 \( 1 \)
11 \( 1 - iT \)
good5 \( 1 - 0.138T + 5T^{2} \)
7 \( 1 + 0.456iT - 7T^{2} \)
13 \( 1 - 5.11iT - 13T^{2} \)
17 \( 1 - 6.17iT - 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 + 9.41T + 23T^{2} \)
29 \( 1 + 7.67T + 29T^{2} \)
31 \( 1 - 5.61iT - 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 - 7.57iT - 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 - 8.10T + 47T^{2} \)
53 \( 1 + 6.90T + 53T^{2} \)
59 \( 1 - 4.46iT - 59T^{2} \)
61 \( 1 - 1.62iT - 61T^{2} \)
67 \( 1 - 4.15T + 67T^{2} \)
71 \( 1 - 1.36T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 + 8.79iT - 79T^{2} \)
83 \( 1 + 7.24iT - 83T^{2} \)
89 \( 1 - 14.4iT - 89T^{2} \)
97 \( 1 - 2.90T + 97T^{2} \)
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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

−10.54897378857526897166807670736, −9.576010034967988321309345230131, −8.959399630439567373633537986308, −7.890814886550001766093555172727, −7.30035268223572300075197497830, −6.21717682737580065162162562035, −5.65177033392756607288818412853, −4.25325592229422426924643339947, −3.81171393635372396576135632542, −1.92780875454666044782247596510, 0.50037756576627639836712829980, 2.17462441781128983018867521762, 3.19549754940971919020886273782, 4.17981786178064307059880764336, 5.46805509418303441984427777402, 5.84661584883493814918344967741, 7.49605694257419820430247597764, 8.230897215446302756521761011577, 9.412973632416095914269680998724, 9.858506795791971824733739837281

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