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Properties

Label 2-1856-116.63-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.831 - 0.556i$
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  + (1.12 + 1.40i)5-s + (0.900 − 0.433i)9-s + (0.400 + 0.193i)13-s − 1.56i·17-s + (−0.500 + 2.19i)25-s + (−0.900 − 0.433i)29-s + (−0.846 − 1.75i)37-s + 1.94i·41-s + (1.62 + 0.781i)45-s + (−0.900 + 0.433i)49-s + (0.777 + 0.974i)53-s + (−1.52 + 0.347i)61-s + (0.178 + 0.781i)65-s + (−1.22 − 0.974i)73-s + (0.623 − 0.781i)81-s + ⋯
L(s)  = 1  + (1.12 + 1.40i)5-s + (0.900 − 0.433i)9-s + (0.400 + 0.193i)13-s − 1.56i·17-s + (−0.500 + 2.19i)25-s + (−0.900 − 0.433i)29-s + (−0.846 − 1.75i)37-s + 1.94i·41-s + (1.62 + 0.781i)45-s + (−0.900 + 0.433i)49-s + (0.777 + 0.974i)53-s + (−1.52 + 0.347i)61-s + (0.178 + 0.781i)65-s + (−1.22 − 0.974i)73-s + (0.623 − 0.781i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.831 - 0.556i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.831 - 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.486395638\)
\(L(\frac12)\) \(\approx\) \(1.486395638\)
\(L(1)\) 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 \( 1 + (0.900 + 0.433i)T \)
good3 \( 1 + (-0.900 + 0.433i)T^{2} \)
5 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
7 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + 1.56iT - T^{2} \)
19 \( 1 + (-0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 - 1.94iT - T^{2} \)
43 \( 1 + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.52 - 0.347i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.678 - 0.541i)T + (0.222 - 0.974i)T^{2} \)
97 \( 1 + (-1.90 - 0.433i)T + (0.900 + 0.433i)T^{2} \)
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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

−9.522999687854317128766316314097, −9.100753162795093839238695031814, −7.57166921199108921652512095009, −7.13797939243258565859676215200, −6.35004741110113099908193736491, −5.72402942673991637900285262353, −4.59510556371748796150432568266, −3.45957252602120579026355410321, −2.63481795421701801842862102178, −1.59541392638387584026884176915, 1.41154473434420367909033173471, 1.95832299758959179497063342199, 3.64280522449154818662132742263, 4.57357790044837703436241612324, 5.32470755081438604563439402752, 5.99679773700418589935732906327, 6.90663191263337374721492211403, 8.030467508170467690404225190785, 8.636985291728799121773288086903, 9.274722089055404951819856060386

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