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Properties

Label 2-244-244.15-c0-0-0
Degree $2$
Conductor $244$
Sign $-0.665 + 0.746i$
Analytic cond. $0.121771$
Root an. cond. $0.348958$
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  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (−1.30 − 1.45i)5-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−1.30 + 1.45i)10-s + (−0.309 + 0.535i)13-s + (0.913 − 0.406i)16-s + (1.58 − 0.336i)17-s + (−0.978 − 0.207i)18-s + (1.58 + 1.14i)20-s + (−0.295 + 2.81i)25-s + (0.564 + 0.251i)26-s + (0.104 + 0.181i)29-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.978 + 0.207i)4-s + (−1.30 − 1.45i)5-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (−1.30 + 1.45i)10-s + (−0.309 + 0.535i)13-s + (0.913 − 0.406i)16-s + (1.58 − 0.336i)17-s + (−0.978 − 0.207i)18-s + (1.58 + 1.14i)20-s + (−0.295 + 2.81i)25-s + (0.564 + 0.251i)26-s + (0.104 + 0.181i)29-s + (−0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 244 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(244\)    =    \(2^{2} \cdot 61\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(0.121771\)
Root analytic conductor: \(0.348958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{244} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 244,\ (\ :0),\ -0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5348338779\)
\(L(\frac12)\) \(\approx\) \(0.5348338779\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
good3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (1.30 + 1.45i)T + (-0.104 + 0.994i)T^{2} \)
7 \( 1 + (-0.669 + 0.743i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \)
19 \( 1 + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.104 - 0.181i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.978 + 0.207i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.913 - 0.406i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.978 - 0.207i)T^{2} \)
67 \( 1 + (0.104 - 0.994i)T^{2} \)
71 \( 1 + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
79 \( 1 + (-0.913 - 0.406i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.139 - 1.33i)T + (-0.978 - 0.207i)T^{2} \)
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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

−12.18696426965100906450900106762, −11.39622233955667330091272054431, −9.954427843220052653034432955458, −9.154995861072454981191777835862, −8.338460228331970158873056958338, −7.37528020879585619313820226129, −5.34433396477285235563137918429, −4.31864238235140234403421413064, −3.44745603713105130267173731296, −1.09615697622952617608287688637, 3.16448234345657177375944974834, 4.34633134076900484221985196370, 5.72764974804262116656979440036, 6.99451955132756718610305685925, 7.68648417606078946317086554133, 8.229949248644840179403496550608, 9.995408607621350072971980485404, 10.54823548592659958113423607108, 11.68391067632260238021777607003, 12.75791906011726759288124941871

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