Properties

Label 2-1792-16.13-c1-0-33
Degree $2$
Conductor $1792$
Sign $0.608 + 0.793i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  + (1.26 + 1.26i)3-s + (−2.95 + 2.95i)5-s i·7-s + 0.189i·9-s + (3.18 − 3.18i)11-s + (−3.42 − 3.42i)13-s − 7.46·15-s − 5.13·17-s + (1.50 + 1.50i)19-s + (1.26 − 1.26i)21-s − 7.11i·23-s − 12.4i·25-s + (3.54 − 3.54i)27-s + (3.84 + 3.84i)29-s − 0.831·31-s + ⋯
L(s)  = 1  + (0.729 + 0.729i)3-s + (−1.32 + 1.32i)5-s − 0.377i·7-s + 0.0630i·9-s + (0.960 − 0.960i)11-s + (−0.950 − 0.950i)13-s − 1.92·15-s − 1.24·17-s + (0.345 + 0.345i)19-s + (0.275 − 0.275i)21-s − 1.48i·23-s − 2.49i·25-s + (0.683 − 0.683i)27-s + (0.714 + 0.714i)29-s − 0.149·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098097960\)
\(L(\frac12)\) \(\approx\) \(1.098097960\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + (-1.26 - 1.26i)T + 3iT^{2} \)
5 \( 1 + (2.95 - 2.95i)T - 5iT^{2} \)
11 \( 1 + (-3.18 + 3.18i)T - 11iT^{2} \)
13 \( 1 + (3.42 + 3.42i)T + 13iT^{2} \)
17 \( 1 + 5.13T + 17T^{2} \)
19 \( 1 + (-1.50 - 1.50i)T + 19iT^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 + (-3.84 - 3.84i)T + 29iT^{2} \)
31 \( 1 + 0.831T + 31T^{2} \)
37 \( 1 + (-5.64 + 5.64i)T - 37iT^{2} \)
41 \( 1 + 2.22iT - 41T^{2} \)
43 \( 1 + (1.61 - 1.61i)T - 43iT^{2} \)
47 \( 1 + 7.83T + 47T^{2} \)
53 \( 1 + (5.58 - 5.58i)T - 53iT^{2} \)
59 \( 1 + (1.85 - 1.85i)T - 59iT^{2} \)
61 \( 1 + (-1.65 - 1.65i)T + 61iT^{2} \)
67 \( 1 + (5.77 + 5.77i)T + 67iT^{2} \)
71 \( 1 - 6.04iT - 71T^{2} \)
73 \( 1 + 7.67iT - 73T^{2} \)
79 \( 1 - 1.90T + 79T^{2} \)
83 \( 1 + (-7.97 - 7.97i)T + 83iT^{2} \)
89 \( 1 - 2.49iT - 89T^{2} \)
97 \( 1 + 1.98T + 97T^{2} \)
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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

−9.076412237415973133212236791598, −8.363570369405549478796125428341, −7.70369604825162265586797394357, −6.81086534728892947822622914711, −6.25898732657426985879830653978, −4.63123436314492042549194454317, −4.00688058668650530314548312698, −3.21708482478231008834113217458, −2.69547755379662813316084566248, −0.38733512793854503681954352524, 1.34406211956486245795723618554, 2.21287704729949985226059942053, 3.55950033757940452240801320775, 4.64456244930987369477138250563, 4.84028957516503782164537961979, 6.51655334351306857377422323060, 7.26457177395652861687955464524, 7.83216156252301280799479996077, 8.558011109953074706522758892700, 9.230007682092722142207689230601

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