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Properties

Label 2-1856-116.71-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.441 - 0.897i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.400 − 0.193i)5-s + (0.222 + 0.974i)9-s + (−0.277 + 1.21i)13-s + 0.867i·17-s + (−0.499 − 0.626i)25-s + (−0.222 + 0.974i)29-s + (1.52 − 0.347i)37-s + 1.56i·41-s + (0.0990 − 0.433i)45-s + (−0.222 − 0.974i)49-s + (1.62 + 0.781i)53-s + (0.678 + 0.541i)61-s + (0.346 − 0.433i)65-s + (−0.376 − 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯
L(s)  = 1  + (−0.400 − 0.193i)5-s + (0.222 + 0.974i)9-s + (−0.277 + 1.21i)13-s + 0.867i·17-s + (−0.499 − 0.626i)25-s + (−0.222 + 0.974i)29-s + (1.52 − 0.347i)37-s + 1.56i·41-s + (0.0990 − 0.433i)45-s + (−0.222 − 0.974i)49-s + (1.62 + 0.781i)53-s + (0.678 + 0.541i)61-s + (0.346 − 0.433i)65-s + (−0.376 − 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.441 - 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9759781007\)
\(L(\frac12)\) \(\approx\) \(0.9759781007\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 - 0.867iT - T^{2} \)
19 \( 1 + (-0.222 + 0.974i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.623 + 0.781i)T^{2} \)
37 \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \)
41 \( 1 - 1.56iT - T^{2} \)
43 \( 1 + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + (-0.900 + 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (-1.22 + 0.974i)T + (0.222 - 0.974i)T^{2} \)
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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

−9.586656604318756660675659741351, −8.680690917630154241629325109630, −8.032275090676452799607818280638, −7.28429392937516010969451124056, −6.46242044458516705699092583169, −5.49842263806496142024307812322, −4.51451505016131034534042735505, −3.98998805816878427026662599927, −2.59581044583718816769976398635, −1.58754385144782791207648641768, 0.75785906488049320529726083398, 2.47320129962803857641586932910, 3.42068950481351453116891051563, 4.21138659531628813221436664589, 5.33410026124228623930235987615, 6.06145940840212701042762342774, 7.07487082890963847756262975541, 7.63152267431950739741544709589, 8.464180545057106500672646560095, 9.436653452355185979337249571950

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