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Properties

Label 2-1792-16.13-c1-0-14
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: $\chi_{1792} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.996258730\)
\(L(\frac12)\) \(\approx\) \(1.996258730\)
\(L(\frac{3}{2})\) 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

−9.842470484174908843518183343240, −8.835109087076610367463096771890, −8.079453138176874799847795308753, −7.72767134455023559111745953975, −6.76733214919980457103963161124, −5.28450336937394562845141153542, −4.58443577094110530783196347432, −3.46538115904165686489062097038, −3.23800079530331507069108330152, −2.30468191067140043984572308211, 0.64250504212589441840089431135, 1.48457104555494894113945998187, 2.82166200675829642751070992941, 3.64939898426812580295661551887, 4.46565167509663957779836172414, 5.75543543279405968061650750863, 6.88540148178818364302194616475, 7.57148604473115228712177634053, 8.038989198635125366375736783394, 8.578485828025091203263362542240

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