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Properties

Label 2-1856-116.91-c0-0-0
Degree $2$
Conductor $1856$
Sign $-0.132 + 0.991i$
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  + (0.277 − 1.21i)5-s + (−0.623 + 0.781i)9-s + (−1.12 − 1.40i)13-s − 1.94i·17-s + (−0.499 − 0.240i)25-s + (0.623 + 0.781i)29-s + (−0.678 − 0.541i)37-s − 0.867i·41-s + (0.777 + 0.974i)45-s + (0.623 − 0.781i)49-s + (0.0990 − 0.433i)53-s + (0.846 + 1.75i)61-s + (−2.02 + 0.974i)65-s + (−1.90 + 0.433i)73-s + (−0.222 − 0.974i)81-s + ⋯
L(s)  = 1  + (0.277 − 1.21i)5-s + (−0.623 + 0.781i)9-s + (−1.12 − 1.40i)13-s − 1.94i·17-s + (−0.499 − 0.240i)25-s + (0.623 + 0.781i)29-s + (−0.678 − 0.541i)37-s − 0.867i·41-s + (0.777 + 0.974i)45-s + (0.623 − 0.781i)49-s + (0.0990 − 0.433i)53-s + (0.846 + 1.75i)61-s + (−2.02 + 0.974i)65-s + (−1.90 + 0.433i)73-s + (−0.222 − 0.974i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $-0.132 + 0.991i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ -0.132 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9561311755\)
\(L(\frac12)\) \(\approx\) \(0.9561311755\)
\(L(1)\) 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 \( 1 + (-0.623 - 0.781i)T \)
good3 \( 1 + (0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
7 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + 1.94iT - T^{2} \)
19 \( 1 + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 + 0.867iT - T^{2} \)
43 \( 1 + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.846 - 1.75i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (1.90 - 0.433i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \)
97 \( 1 + (-0.376 + 0.781i)T + (-0.623 - 0.781i)T^{2} \)
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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

−9.036956393296275551264426867662, −8.605833436008037557183063202551, −7.64275793312337879027425073318, −7.08846755434206876470589260683, −5.54162139504240658744795163842, −5.26934838450220063745765022654, −4.61229113756574077795230522595, −3.10084719508082208512993223552, −2.27861134113875725267814656742, −0.68714177442739678459872638919, 1.85581136262045463144989273130, 2.79751305334716412437464171858, 3.73071833504309634857705457389, 4.64680144216594943089479333351, 5.99365890763665698155157839361, 6.42071600200460360638862081476, 7.09475854105376057965522271939, 8.081238219061054046662020328251, 8.920573201509133976222009844045, 9.723126854044909839927512121590

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