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Properties

Label 2-1792-16.13-c1-0-47
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  + (−0.171 − 0.171i)3-s + (0.268 − 0.268i)5-s − i·7-s − 2.94i·9-s + (1.84 − 1.84i)11-s + (−1.63 − 1.63i)13-s − 0.0919·15-s − 7.37·17-s + (−3.84 − 3.84i)19-s + (−0.171 + 0.171i)21-s + 6.44i·23-s + 4.85i·25-s + (−1.01 + 1.01i)27-s + (−3.58 − 3.58i)29-s + 6.10·31-s + ⋯
L(s)  = 1  + (−0.0988 − 0.0988i)3-s + (0.120 − 0.120i)5-s − 0.377i·7-s − 0.980i·9-s + (0.557 − 0.557i)11-s + (−0.453 − 0.453i)13-s − 0.0237·15-s − 1.78·17-s + (−0.883 − 0.883i)19-s + (−0.0373 + 0.0373i)21-s + 1.34i·23-s + 0.971i·25-s + (−0.195 + 0.195i)27-s + (−0.666 − 0.666i)29-s + 1.09·31-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

\(L(1)\) \(\approx\) \(0.6007771193\)
\(L(\frac12)\) \(\approx\) \(0.6007771193\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.171 + 0.171i)T + 3iT^{2} \)
5 \( 1 + (-0.268 + 0.268i)T - 5iT^{2} \)
11 \( 1 + (-1.84 + 1.84i)T - 11iT^{2} \)
13 \( 1 + (1.63 + 1.63i)T + 13iT^{2} \)
17 \( 1 + 7.37T + 17T^{2} \)
19 \( 1 + (3.84 + 3.84i)T + 19iT^{2} \)
23 \( 1 - 6.44iT - 23T^{2} \)
29 \( 1 + (3.58 + 3.58i)T + 29iT^{2} \)
31 \( 1 - 6.10T + 31T^{2} \)
37 \( 1 + (7.41 - 7.41i)T - 37iT^{2} \)
41 \( 1 + 0.836iT - 41T^{2} \)
43 \( 1 + (3.88 - 3.88i)T - 43iT^{2} \)
47 \( 1 - 6.02T + 47T^{2} \)
53 \( 1 + (0.575 - 0.575i)T - 53iT^{2} \)
59 \( 1 + (-5.33 + 5.33i)T - 59iT^{2} \)
61 \( 1 + (0.929 + 0.929i)T + 61iT^{2} \)
67 \( 1 + (6.21 + 6.21i)T + 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + 3.68iT - 73T^{2} \)
79 \( 1 - 4.21T + 79T^{2} \)
83 \( 1 + (12.0 + 12.0i)T + 83iT^{2} \)
89 \( 1 + 9.32iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.997534502967853217524700722074, −8.236813696547013049241514405457, −7.08513938809651032672157902208, −6.62469956690943247057759886352, −5.77209324140631566955449901318, −4.71897269316865428767275312632, −3.87792479467252118360385855565, −2.92124204703028578084348988228, −1.57469103726827047701056724463, −0.21190087973147514510255452532, 1.96931233585519119272441813329, 2.45076388097079676850567837375, 4.14280420884451812469110488359, 4.57281613527954802230481762407, 5.62797687976421466278330806561, 6.60945065483200229331494079791, 7.07786474812498391688548674414, 8.327282983507286068051581923165, 8.727780842798804625496086240525, 9.662116432133006691135424393618

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