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Properties

Label 2-1856-116.103-c0-0-0
Degree $2$
Conductor $1856$
Sign $-0.128 + 0.991i$
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  + (−0.400 − 0.193i)5-s + (−0.222 − 0.974i)9-s + (0.277 − 1.21i)13-s − 1.80·17-s + (−0.499 − 0.626i)25-s + (0.222 − 0.974i)29-s + (0.277 + 1.21i)37-s + 1.24·41-s + (−0.0990 + 0.433i)45-s + (−0.222 − 0.974i)49-s + (−1.62 − 0.781i)53-s + (1.12 − 1.40i)61-s + (−0.346 + 0.433i)65-s + (1.62 − 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 − 1.80·17-s + (−0.499 − 0.626i)25-s + (0.222 − 0.974i)29-s + (0.277 + 1.21i)37-s + 1.24·41-s + (−0.0990 + 0.433i)45-s + (−0.222 − 0.974i)49-s + (−1.62 − 0.781i)53-s + (1.12 − 1.40i)61-s + (−0.346 + 0.433i)65-s + (1.62 − 0.781i)73-s + (−0.900 + 0.433i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ -0.128 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8123455558\)
\(L(\frac12)\) \(\approx\) \(0.8123455558\)
\(L(1)\) 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 \( 1 + 1.80T + T^{2} \)
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 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 - 1.24T + T^{2} \)
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 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
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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

−9.224153490808330227969199212526, −8.307541876976729492510455874480, −7.911188906439233124124803556554, −6.61102723005756695648545650756, −6.24761742846636720498872900335, −5.10122894579329939120263907821, −4.20499878698334885868560809143, −3.37038309719593835125073094183, −2.28867404281208537476563313998, −0.58715733322287378105171032627, 1.78483012795885437338673264181, 2.69762829553008996335439194140, 4.02191864113879803354383958044, 4.58485122340742873241838176473, 5.63595999643034259112255141115, 6.59447919041054108108118647104, 7.26735877261215996079141156591, 8.057528753777896248630747297479, 8.950943115007874152239649597676, 9.397923947647493160072990495733

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