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Properties

Label 2-226-113.112-c1-0-8
Degree $2$
Conductor $226$
Sign $-0.846 + 0.532i$
Analytic cond. $1.80461$
Root an. cond. $1.34336$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.41i·3-s + 4-s − 2.82i·5-s + 1.41i·6-s − 4·7-s − 8-s + 0.999·9-s + 2.82i·10-s − 1.41i·12-s − 4·13-s + 4·14-s − 4.00·15-s + 16-s + 5.65i·17-s − 0.999·18-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.816i·3-s + 0.5·4-s − 1.26i·5-s + 0.577i·6-s − 1.51·7-s − 0.353·8-s + 0.333·9-s + 0.894i·10-s − 0.408i·12-s − 1.10·13-s + 1.06·14-s − 1.03·15-s + 0.250·16-s + 1.37i·17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 226 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(226\)    =    \(2 \cdot 113\)
Sign: $-0.846 + 0.532i$
Analytic conductor: \(1.80461\)
Root analytic conductor: \(1.34336\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{226} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 226,\ (\ :1/2),\ -0.846 + 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166438 - 0.577568i\)
\(L(\frac12)\) \(\approx\) \(0.166438 - 0.577568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
113 \( 1 + (-9 + 5.65i)T \)
good3 \( 1 + 1.41iT - 3T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 + 7.07iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 - 9.89iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + 7.07iT - 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.31141079418763191491288505633, −10.70096008605305733963445737444, −9.638314203437256002275773930041, −9.001857723805051246630429721182, −7.87234642956242869761994818953, −6.89257024341965846620786518238, −5.96594262407378662303220721319, −4.31394680844603180552755485570, −2.37442649096191884035668307047, −0.60205853275041224072904327123, 2.74714091808275711222503104768, 3.69535666331519582135264799592, 5.54515067934607623186954799462, 6.95339628049080709813870601373, 7.32351798829068601037656538487, 9.194661205689948471468017094572, 9.910540253202443326284011209513, 10.24724742728993739816746233272, 11.40221014391323909181777223038, 12.42044212405947585736859845485

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