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Properties

Label 2-792-11.4-c1-0-14
Degree $2$
Conductor $792$
Sign $-0.908 + 0.417i$
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  + (0.309 − 0.224i)5-s + (−1.30 − 4.02i)7-s + (−3.04 + 1.31i)11-s + (−1.80 − 1.31i)13-s + (−1.5 + 1.08i)17-s + (0.118 − 0.363i)19-s − 6.23·23-s + (−1.5 + 4.61i)25-s + (−0.145 − 0.449i)29-s + (−6.97 − 5.06i)31-s + (−1.30 − 0.951i)35-s + (1.16 + 3.57i)37-s + (1.54 − 4.75i)41-s + 11.4·43-s + (−0.0450 + 0.138i)47-s + ⋯
L(s)  = 1  + (0.138 − 0.100i)5-s + (−0.494 − 1.52i)7-s + (−0.918 + 0.396i)11-s + (−0.501 − 0.364i)13-s + (−0.363 + 0.264i)17-s + (0.0270 − 0.0833i)19-s − 1.30·23-s + (−0.300 + 0.923i)25-s + (−0.0270 − 0.0833i)29-s + (−1.25 − 0.909i)31-s + (−0.221 − 0.160i)35-s + (0.191 + 0.588i)37-s + (0.241 − 0.742i)41-s + 1.74·43-s + (−0.00657 + 0.0202i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $-0.908 + 0.417i$
Analytic conductor: \(6.32415\)
Root analytic conductor: \(2.51478\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 792,\ (\ :1/2),\ -0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122154 - 0.557717i\)
\(L(\frac12)\) \(\approx\) \(0.122154 - 0.557717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 - 1.31i)T \)
good5 \( 1 + (-0.309 + 0.224i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.30 + 4.02i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.80 + 1.31i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.5 - 1.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.118 + 0.363i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 + (0.145 + 0.449i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.97 + 5.06i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.16 - 3.57i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.54 + 4.75i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (0.0450 - 0.138i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.73 + 5.62i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.5 + 13.8i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.54 + 4.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.56T + 67T^{2} \)
71 \( 1 + (-10.1 + 7.38i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.14 - 3.52i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.5 + 8.36i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.57 - 1.14i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 3.47T + 89T^{2} \)
97 \( 1 + (-5.54 - 4.02i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.970388200686257643099383341253, −9.314207124438456288878753860850, −7.84468249575400652402312034349, −7.53460203715226403192867964813, −6.48862466649686905662604999192, −5.43437171188260895405462592204, −4.35655100658539923520688740330, −3.48169877732713533332056752804, −2.04992045875617772577434027870, −0.26089921027989525029643704668, 2.18527742640660848166619280353, 2.90513532703903446328577605376, 4.35973241215345589689360605096, 5.59553559887565534769689252355, 6.01018157444758270289247381205, 7.21484592145938658404428756833, 8.202491373975087728492033878091, 9.006793680626546642250560958792, 9.690704505358796483445559284752, 10.60135638292716750434009164172

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