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Properties

Label 2-1792-16.5-c1-0-31
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  + (1.48 − 1.48i)3-s + (1.83 + 1.83i)5-s + i·7-s − 1.40i·9-s + (0.321 + 0.321i)11-s + (4.61 − 4.61i)13-s + 5.45·15-s − 1.84·17-s + (−3.88 + 3.88i)19-s + (1.48 + 1.48i)21-s + 5.88i·23-s + 1.74i·25-s + (2.36 + 2.36i)27-s + (6.14 − 6.14i)29-s + 5.69·31-s + ⋯
L(s)  = 1  + (0.857 − 0.857i)3-s + (0.821 + 0.821i)5-s + 0.377i·7-s − 0.469i·9-s + (0.0969 + 0.0969i)11-s + (1.28 − 1.28i)13-s + 1.40·15-s − 0.446·17-s + (−0.892 + 0.892i)19-s + (0.323 + 0.323i)21-s + 1.22i·23-s + 0.348i·25-s + (0.454 + 0.454i)27-s + (1.14 − 1.14i)29-s + 1.02·31-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: $\chi_{1792} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.892485883\)
\(L(\frac12)\) \(\approx\) \(2.892485883\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 + 1.48i)T - 3iT^{2} \)
5 \( 1 + (-1.83 - 1.83i)T + 5iT^{2} \)
11 \( 1 + (-0.321 - 0.321i)T + 11iT^{2} \)
13 \( 1 + (-4.61 + 4.61i)T - 13iT^{2} \)
17 \( 1 + 1.84T + 17T^{2} \)
19 \( 1 + (3.88 - 3.88i)T - 19iT^{2} \)
23 \( 1 - 5.88iT - 23T^{2} \)
29 \( 1 + (-6.14 + 6.14i)T - 29iT^{2} \)
31 \( 1 - 5.69T + 31T^{2} \)
37 \( 1 + (-1.66 - 1.66i)T + 37iT^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + (0.533 + 0.533i)T + 43iT^{2} \)
47 \( 1 + 0.465T + 47T^{2} \)
53 \( 1 + (0.623 + 0.623i)T + 53iT^{2} \)
59 \( 1 + (-7.32 - 7.32i)T + 59iT^{2} \)
61 \( 1 + (7.57 - 7.57i)T - 61iT^{2} \)
67 \( 1 + (6.16 - 6.16i)T - 67iT^{2} \)
71 \( 1 + 0.162iT - 71T^{2} \)
73 \( 1 - 3.49iT - 73T^{2} \)
79 \( 1 - 8.28T + 79T^{2} \)
83 \( 1 + (2.51 - 2.51i)T - 83iT^{2} \)
89 \( 1 - 1.60iT - 89T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−9.059174274523049703564570359252, −8.345919617870324597930508293042, −7.85703216144743287550338058551, −6.84676211320965026962458150997, −6.15170724299984461377923013934, −5.56266466545950328613262243624, −4.02587101337382778573397970266, −2.96021301996105356714106758545, −2.34767011651383954953541565227, −1.32422790825944090516080392374, 1.17104199250749510773973423145, 2.36092924837312267071923668441, 3.44899257655495032567134432239, 4.55407169947515226325753610483, 4.69998811712164662113044704083, 6.38628050716345892351240261166, 6.53860559751102514697745315329, 8.181377921591313277511722883632, 8.780543715117250489725510386631, 9.102642546206719802941267703523

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