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Properties

Label 2-1792-16.5-c1-0-10
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  + (−1.26 + 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

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

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

\(L(1)\) \(\approx\) \(0.9492141067\)
\(L(\frac12)\) \(\approx\) \(0.9492141067\)
\(L(\frac{3}{2})\) 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

−9.900013112416010970291823948348, −9.112575799917745666816625553160, −8.168771419036755664778531464977, −7.14877180466442683111009466075, −6.15384921850716186816075911977, −5.66400947368433494386174953884, −5.24021197766554445959610962788, −3.69982391958596636661183466607, −2.89004474038949886633381455572, −1.87474941768895678449254964207, 0.35856906070082430401483563515, 1.63087195578963084039156341582, 2.23425971046131686668181412492, 4.21294301441118705573790910511, 4.86652544442032103554970637075, 5.68353480654021425653084989048, 6.53855037687944615098382799370, 6.89882863359796511858339903156, 8.260823299208740008071206593968, 8.859175542141613245195905702134

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