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Properties

Label 2-1792-16.13-c1-0-36
Degree $2$
Conductor $1792$
Sign $0.991 - 0.130i$
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  + (1.62 + 1.62i)3-s + (1.16 − 1.16i)5-s + i·7-s + 2.28i·9-s + (4.39 − 4.39i)11-s + (−0.448 − 0.448i)13-s + 3.80·15-s + 5.02·17-s + (−1.49 − 1.49i)19-s + (−1.62 + 1.62i)21-s − 8.89i·23-s + 2.26i·25-s + (1.15 − 1.15i)27-s + (0.803 + 0.803i)29-s − 8.27·31-s + ⋯
L(s)  = 1  + (0.938 + 0.938i)3-s + (0.522 − 0.522i)5-s + 0.377i·7-s + 0.762i·9-s + (1.32 − 1.32i)11-s + (−0.124 − 0.124i)13-s + 0.981·15-s + 1.21·17-s + (−0.343 − 0.343i)19-s + (−0.354 + 0.354i)21-s − 1.85i·23-s + 0.453i·25-s + (0.222 − 0.222i)27-s + (0.149 + 0.149i)29-s − 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.991 - 0.130i$
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: \((2,\ 1792,\ (\ :1/2),\ 0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.904635293\)
\(L(\frac12)\) \(\approx\) \(2.904635293\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 1.62i)T + 3iT^{2} \)
5 \( 1 + (-1.16 + 1.16i)T - 5iT^{2} \)
11 \( 1 + (-4.39 + 4.39i)T - 11iT^{2} \)
13 \( 1 + (0.448 + 0.448i)T + 13iT^{2} \)
17 \( 1 - 5.02T + 17T^{2} \)
19 \( 1 + (1.49 + 1.49i)T + 19iT^{2} \)
23 \( 1 + 8.89iT - 23T^{2} \)
29 \( 1 + (-0.803 - 0.803i)T + 29iT^{2} \)
31 \( 1 + 8.27T + 31T^{2} \)
37 \( 1 + (1.55 - 1.55i)T - 37iT^{2} \)
41 \( 1 + 4.93iT - 41T^{2} \)
43 \( 1 + (3.73 - 3.73i)T - 43iT^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 + (4.53 - 4.53i)T - 53iT^{2} \)
59 \( 1 + (-1.06 + 1.06i)T - 59iT^{2} \)
61 \( 1 + (-5.11 - 5.11i)T + 61iT^{2} \)
67 \( 1 + (-7.47 - 7.47i)T + 67iT^{2} \)
71 \( 1 + 4.07iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + 9.19T + 79T^{2} \)
83 \( 1 + (-2.62 - 2.62i)T + 83iT^{2} \)
89 \( 1 + 1.60iT - 89T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−9.052895248678487945395966540956, −8.829305171668245468635899827027, −8.143826885549172296927770482477, −6.86038159220765661580187074114, −5.93436548267844536530067656451, −5.20243440248512994117785917094, −4.12069744914380372175846853967, −3.45212729519161344092122765831, −2.52750547136771935570696399469, −1.10157726088989347253225544670, 1.50362963356739495917979138300, 1.97477656845543215840880121911, 3.24302098890327350165853293243, 3.99601196773337899390970439899, 5.30162044299073831758170628483, 6.35172519540732370244661789362, 7.11516668394272675312415416112, 7.48947446248991092880786391130, 8.349970642878516190285675436463, 9.448570527230783132420701164924

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