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Properties

Label 2-73-73.27-c1-0-0
Degree $2$
Conductor $73$
Sign $0.916 - 0.399i$
Analytic cond. $0.582907$
Root an. cond. $0.763484$
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  − 2.22·2-s + 0.185i·3-s + 2.96·4-s + (−0.161 − 0.161i)5-s − 0.414i·6-s + (2.85 + 2.85i)7-s − 2.15·8-s + 2.96·9-s + (0.359 + 0.359i)10-s + (0.545 − 0.545i)11-s + 0.551i·12-s + (−1.51 + 1.51i)13-s + (−6.36 − 6.36i)14-s + (0.0300 − 0.0300i)15-s − 1.13·16-s + (−2.30 − 2.30i)17-s + ⋯
L(s)  = 1  − 1.57·2-s + 0.107i·3-s + 1.48·4-s + (−0.0722 − 0.0722i)5-s − 0.169i·6-s + (1.08 + 1.08i)7-s − 0.760·8-s + 0.988·9-s + (0.113 + 0.113i)10-s + (0.164 − 0.164i)11-s + 0.159i·12-s + (−0.419 + 0.419i)13-s + (−1.70 − 1.70i)14-s + (0.00774 − 0.00774i)15-s − 0.284·16-s + (−0.559 − 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(73\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(0.582907\)
Root analytic conductor: \(0.763484\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{73} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 73,\ (\ :1/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508553 + 0.105922i\)
\(L(\frac12)\) \(\approx\) \(0.508553 + 0.105922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad73 \( 1 + (8.38 + 1.65i)T \)
good2 \( 1 + 2.22T + 2T^{2} \)
3 \( 1 - 0.185iT - 3T^{2} \)
5 \( 1 + (0.161 + 0.161i)T + 5iT^{2} \)
7 \( 1 + (-2.85 - 2.85i)T + 7iT^{2} \)
11 \( 1 + (-0.545 + 0.545i)T - 11iT^{2} \)
13 \( 1 + (1.51 - 1.51i)T - 13iT^{2} \)
17 \( 1 + (2.30 + 2.30i)T + 17iT^{2} \)
19 \( 1 + 2.93iT - 19T^{2} \)
23 \( 1 - 6.74iT - 23T^{2} \)
29 \( 1 + (1.85 - 1.85i)T - 29iT^{2} \)
31 \( 1 + (-5.51 + 5.51i)T - 31iT^{2} \)
37 \( 1 + 7.25T + 37T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 + (6.76 - 6.76i)T - 43iT^{2} \)
47 \( 1 + (-8.86 + 8.86i)T - 47iT^{2} \)
53 \( 1 + (4.57 + 4.57i)T + 53iT^{2} \)
59 \( 1 + (0.143 + 0.143i)T + 59iT^{2} \)
61 \( 1 + 11.5iT - 61T^{2} \)
67 \( 1 + 6.91iT - 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + (-8.38 - 8.38i)T + 83iT^{2} \)
89 \( 1 + 4.13T + 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−15.23011988745284392786634945248, −13.65955720936190152696112183687, −11.96427794282642576696796046532, −11.23617784713365294905343267266, −9.901029853784607309157910287825, −9.017578273031543307539718345326, −8.037692158131084627403283016499, −6.86666659799699820494140382754, −4.86075679388021621148540317243, −1.94264159416704706938733525424, 1.51786692521295267394449461153, 4.46046090895153177068268297472, 6.90339907513652667089263548902, 7.69660571864435868398857399980, 8.729009651850204050085462963574, 10.30191034455930880444120518397, 10.57451259228679081681431364199, 12.02179593432206160554669971502, 13.48305463359362655721817252261, 14.78278368562394463814766155640

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