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Properties

Label 2-1856-29.8-c0-0-0
Degree $2$
Conductor $1856$
Sign $-0.191 + 0.981i$
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  + (−0.846 − 1.75i)5-s + (0.974 + 0.222i)9-s + (1.21 − 0.277i)13-s + (−1.33 − 1.33i)17-s + (−1.74 + 2.19i)25-s + (−0.974 + 0.222i)29-s + (−0.119 − 0.189i)37-s + (1.40 − 1.40i)41-s + (−0.433 − 1.90i)45-s + (0.222 − 0.974i)49-s + (−0.781 + 0.376i)53-s + (0.0739 − 0.656i)61-s + (−1.51 − 1.90i)65-s + (−0.623 + 0.218i)73-s + (0.900 + 0.433i)81-s + ⋯
L(s)  = 1  + (−0.846 − 1.75i)5-s + (0.974 + 0.222i)9-s + (1.21 − 0.277i)13-s + (−1.33 − 1.33i)17-s + (−1.74 + 2.19i)25-s + (−0.974 + 0.222i)29-s + (−0.119 − 0.189i)37-s + (1.40 − 1.40i)41-s + (−0.433 − 1.90i)45-s + (0.222 − 0.974i)49-s + (−0.781 + 0.376i)53-s + (0.0739 − 0.656i)61-s + (−1.51 − 1.90i)65-s + (−0.623 + 0.218i)73-s + (0.900 + 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $-0.191 + 0.981i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ -0.191 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9688937766\)
\(L(\frac12)\) \(\approx\) \(0.9688937766\)
\(L(1)\) 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 \( 1 + (0.974 - 0.222i)T \)
good3 \( 1 + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.433 - 0.900i)T^{2} \)
13 \( 1 + (-1.21 + 0.277i)T + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (1.33 + 1.33i)T + iT^{2} \)
19 \( 1 + (0.974 - 0.222i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.781 + 0.623i)T^{2} \)
37 \( 1 + (0.119 + 0.189i)T + (-0.433 + 0.900i)T^{2} \)
41 \( 1 + (-1.40 + 1.40i)T - iT^{2} \)
43 \( 1 + (-0.781 + 0.623i)T^{2} \)
47 \( 1 + (0.433 + 0.900i)T^{2} \)
53 \( 1 + (0.781 - 0.376i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.0739 + 0.656i)T + (-0.974 - 0.222i)T^{2} \)
67 \( 1 + (0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.623 - 0.218i)T + (0.781 - 0.623i)T^{2} \)
79 \( 1 + (0.433 - 0.900i)T^{2} \)
83 \( 1 + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.0250 + 0.222i)T + (-0.974 + 0.222i)T^{2} \)
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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

−9.007670589031456313957171383381, −8.612836423170161545593251223355, −7.65877638023020966758601776564, −7.09482811788526336023345358806, −5.82993177489178407673651432706, −4.95968091970748620454876927398, −4.33131504640453436932125398557, −3.62429245564968390183386094224, −1.92109839288558843677653748746, −0.75725998569225538487411202036, 1.76662454808076137960852816448, 2.96252007483634290308796631557, 3.94620019596035055895335629518, 4.26973758519600455304476834657, 6.13618050822569623283081776011, 6.42146333195351600493322856176, 7.27292476459713157861968620485, 7.910549201235445467719903174485, 8.793658540042246511712233616297, 9.799590037953804823479026857794

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