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Properties

Label 2-273-39.5-c1-0-20
Degree $2$
Conductor $273$
Sign $0.408 + 0.912i$
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  + (−0.0762 − 0.0762i)2-s + (1.58 + 0.701i)3-s − 1.98i·4-s + (−2.41 − 2.41i)5-s + (−0.0672 − 0.174i)6-s + (0.707 + 0.707i)7-s + (−0.303 + 0.303i)8-s + (2.01 + 2.22i)9-s + 0.368i·10-s + (3.75 − 3.75i)11-s + (1.39 − 3.14i)12-s + (−3.50 − 0.861i)13-s − 0.107i·14-s + (−2.13 − 5.52i)15-s − 3.93·16-s − 0.501·17-s + ⋯
L(s)  = 1  + (−0.0538 − 0.0538i)2-s + (0.914 + 0.405i)3-s − 0.994i·4-s + (−1.08 − 1.08i)5-s + (−0.0274 − 0.0711i)6-s + (0.267 + 0.267i)7-s + (−0.107 + 0.107i)8-s + (0.671 + 0.740i)9-s + 0.116i·10-s + (1.13 − 1.13i)11-s + (0.402 − 0.908i)12-s + (−0.971 − 0.238i)13-s − 0.0288i·14-s + (−0.550 − 1.42i)15-s − 0.982·16-s − 0.121·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.408 + 0.912i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20328 - 0.779538i\)
\(L(\frac12)\) \(\approx\) \(1.20328 - 0.779538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 - 0.701i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (3.50 + 0.861i)T \)
good2 \( 1 + (0.0762 + 0.0762i)T + 2iT^{2} \)
5 \( 1 + (2.41 + 2.41i)T + 5iT^{2} \)
11 \( 1 + (-3.75 + 3.75i)T - 11iT^{2} \)
17 \( 1 + 0.501T + 17T^{2} \)
19 \( 1 + (-3.52 + 3.52i)T - 19iT^{2} \)
23 \( 1 - 4.77T + 23T^{2} \)
29 \( 1 - 6.76iT - 29T^{2} \)
31 \( 1 + (3.11 - 3.11i)T - 31iT^{2} \)
37 \( 1 + (-2.59 - 2.59i)T + 37iT^{2} \)
41 \( 1 + (-1.35 - 1.35i)T + 41iT^{2} \)
43 \( 1 - 5.63iT - 43T^{2} \)
47 \( 1 + (4.05 - 4.05i)T - 47iT^{2} \)
53 \( 1 - 4.05iT - 53T^{2} \)
59 \( 1 + (-9.92 + 9.92i)T - 59iT^{2} \)
61 \( 1 - 0.198T + 61T^{2} \)
67 \( 1 + (2.62 - 2.62i)T - 67iT^{2} \)
71 \( 1 + (5.51 + 5.51i)T + 71iT^{2} \)
73 \( 1 + (5.30 + 5.30i)T + 73iT^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + (-4.32 - 4.32i)T + 83iT^{2} \)
89 \( 1 + (1.57 - 1.57i)T - 89iT^{2} \)
97 \( 1 + (4.63 - 4.63i)T - 97iT^{2} \)
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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

−11.57323168915846816192613889194, −10.88199284335112582857356449028, −9.416903095226435673218785592373, −9.036531632441775766338326565805, −8.147552589575270236421586872743, −6.95618167297160911798175235667, −5.24029783003319462556833129282, −4.56077202954493457298671447859, −3.16488269526169417053058400363, −1.16570638472116653703641291895, 2.35576622862907332236092444395, 3.60804588577471303924898857684, 4.26629413836056634727819050076, 6.83537875017583835353915620708, 7.30782412408951907397078420427, 7.87615437828588475103102379280, 9.076414251317754287157449153219, 10.01724020104140659000863494539, 11.57045890460812687711635601875, 11.94215481219931956626941795999

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