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Properties

Label 2-1792-16.5-c1-0-2
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  + (−2.41 + 2.41i)3-s + (−2.54 − 2.54i)5-s + i·7-s − 8.70i·9-s + (0.764 + 0.764i)11-s + (−1.26 + 1.26i)13-s + 12.2·15-s + 5.65·17-s + (0.0445 − 0.0445i)19-s + (−2.41 − 2.41i)21-s − 1.46i·23-s + 7.91i·25-s + (13.8 + 13.8i)27-s + (3.56 − 3.56i)29-s − 4.75·31-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)3-s + (−1.13 − 1.13i)5-s + 0.377i·7-s − 2.90i·9-s + (0.230 + 0.230i)11-s + (−0.351 + 0.351i)13-s + 3.17·15-s + 1.37·17-s + (0.0102 − 0.0102i)19-s + (−0.527 − 0.527i)21-s − 0.305i·23-s + 1.58i·25-s + (2.65 + 2.65i)27-s + (0.662 − 0.662i)29-s − 0.853·31-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

\(L(1)\) \(\approx\) \(0.2413313041\)
\(L(\frac12)\) \(\approx\) \(0.2413313041\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 - 2.41i)T - 3iT^{2} \)
5 \( 1 + (2.54 + 2.54i)T + 5iT^{2} \)
11 \( 1 + (-0.764 - 0.764i)T + 11iT^{2} \)
13 \( 1 + (1.26 - 1.26i)T - 13iT^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
19 \( 1 + (-0.0445 + 0.0445i)T - 19iT^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 + (-3.56 + 3.56i)T - 29iT^{2} \)
31 \( 1 + 4.75T + 31T^{2} \)
37 \( 1 + (5.09 + 5.09i)T + 37iT^{2} \)
41 \( 1 - 7.50iT - 41T^{2} \)
43 \( 1 + (-3.22 - 3.22i)T + 43iT^{2} \)
47 \( 1 - 1.52T + 47T^{2} \)
53 \( 1 + (-4.66 - 4.66i)T + 53iT^{2} \)
59 \( 1 + (5.38 + 5.38i)T + 59iT^{2} \)
61 \( 1 + (6.80 - 6.80i)T - 61iT^{2} \)
67 \( 1 + (-4.92 + 4.92i)T - 67iT^{2} \)
71 \( 1 + 6.19iT - 71T^{2} \)
73 \( 1 + 8.59iT - 73T^{2} \)
79 \( 1 + 7.84T + 79T^{2} \)
83 \( 1 + (7.43 - 7.43i)T - 83iT^{2} \)
89 \( 1 - 9.32iT - 89T^{2} \)
97 \( 1 - 0.485T + 97T^{2} \)
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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

−9.570346862496095268662248363348, −9.164100877581024723426498077564, −8.237134359345250246617124579864, −7.25696058746116946878774486049, −6.13245803776104785440057629520, −5.40403436217427491473535225758, −4.70641949787028689250490562575, −4.16863650793074924695804812495, −3.32696386664750911283196183222, −1.03607458072999860069160205750, 0.14537420343080916714299181014, 1.35070606048408755346248771963, 2.77347531416320017836560494779, 3.80500318370391442079873908690, 5.08676714483651905717082426179, 5.81161363481292972738406164012, 6.73419672087798777664485295738, 7.27735375631494052966842428028, 7.64359576385299301883006198583, 8.494085210930317319319390387282

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