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Properties

Label 2-273-13.12-c1-0-6
Degree $2$
Conductor $273$
Sign $0.911 + 0.410i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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  − 1.48i·2-s + 3-s − 0.193·4-s + 4.15i·5-s − 1.48i·6-s + i·7-s − 2.67i·8-s + 9-s + 6.15·10-s + 3.19i·11-s − 0.193·12-s + (1.48 − 3.28i)13-s + 1.48·14-s + 4.15i·15-s − 4.35·16-s + 3.35·17-s + ⋯
L(s)  = 1  − 1.04i·2-s + 0.577·3-s − 0.0969·4-s + 1.85i·5-s − 0.604i·6-s + 0.377i·7-s − 0.945i·8-s + 0.333·9-s + 1.94·10-s + 0.963i·11-s − 0.0559·12-s + (0.410 − 0.911i)13-s + 0.395·14-s + 1.07i·15-s − 1.08·16-s + 0.812·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.911 + 0.410i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62783 - 0.349804i\)
\(L(\frac12)\) \(\approx\) \(1.62783 - 0.349804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - iT \)
13 \( 1 + (-1.48 + 3.28i)T \)
good2 \( 1 + 1.48iT - 2T^{2} \)
5 \( 1 - 4.15iT - 5T^{2} \)
11 \( 1 - 3.19iT - 11T^{2} \)
17 \( 1 - 3.35T + 17T^{2} \)
19 \( 1 + 2.38iT - 19T^{2} \)
23 \( 1 - 0.387T + 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 - 1.92T + 43T^{2} \)
47 \( 1 - 3.76iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6.15iT - 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 5.61iT - 67T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 + 8.96T + 79T^{2} \)
83 \( 1 + 6.99iT - 83T^{2} \)
89 \( 1 + 0.932iT - 89T^{2} \)
97 \( 1 - 3.35iT - 97T^{2} \)
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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

−11.56720476208974907813891072134, −10.94397564747974915950440206639, −10.06455916659142812126974281203, −9.516988883160988772998483656947, −7.77011228778174988486872789695, −7.09291363604466615922813667379, −5.94391882187187924690171256339, −3.87614953088857811018814751358, −2.96676757074889381464756178226, −2.14084616359296801334897363508, 1.52575099620419627149295276262, 3.76025364319328962582934486898, 5.03503493557321978763522096659, 5.88370679230778932494542565513, 7.19932835911843331415049322696, 8.273983413923977410220968786711, 8.662685200341064569525488039054, 9.623737152634746465591811302880, 11.13230041507178718147523479814, 12.13448636713365840681108619974

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