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Properties

Label 2-273-91.83-c1-0-14
Degree $2$
Conductor $273$
Sign $0.432 + 0.901i$
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  + (1.71 − 1.71i)2-s + i·3-s − 3.91i·4-s + (0.968 + 0.968i)5-s + (1.71 + 1.71i)6-s + (2.61 + 0.407i)7-s + (−3.29 − 3.29i)8-s − 9-s + 3.32·10-s + (−1.56 − 1.56i)11-s + 3.91·12-s + (−3.43 + 1.09i)13-s + (5.19 − 3.79i)14-s + (−0.968 + 0.968i)15-s − 3.49·16-s − 0.815·17-s + ⋯
L(s)  = 1  + (1.21 − 1.21i)2-s + 0.577i·3-s − 1.95i·4-s + (0.432 + 0.432i)5-s + (0.702 + 0.702i)6-s + (0.988 + 0.154i)7-s + (−1.16 − 1.16i)8-s − 0.333·9-s + 1.05·10-s + (−0.470 − 0.470i)11-s + 1.13·12-s + (−0.952 + 0.303i)13-s + (1.38 − 1.01i)14-s + (−0.249 + 0.249i)15-s − 0.873·16-s − 0.197·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.432 + 0.901i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.432 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07686 - 1.30653i\)
\(L(\frac12)\) \(\approx\) \(2.07686 - 1.30653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-2.61 - 0.407i)T \)
13 \( 1 + (3.43 - 1.09i)T \)
good2 \( 1 + (-1.71 + 1.71i)T - 2iT^{2} \)
5 \( 1 + (-0.968 - 0.968i)T + 5iT^{2} \)
11 \( 1 + (1.56 + 1.56i)T + 11iT^{2} \)
17 \( 1 + 0.815T + 17T^{2} \)
19 \( 1 + (3.54 + 3.54i)T + 19iT^{2} \)
23 \( 1 - 6.45iT - 23T^{2} \)
29 \( 1 + 6.34T + 29T^{2} \)
31 \( 1 + (-1.93 - 1.93i)T + 31iT^{2} \)
37 \( 1 + (-5.45 - 5.45i)T + 37iT^{2} \)
41 \( 1 + (-2.86 - 2.86i)T + 41iT^{2} \)
43 \( 1 - 1.41iT - 43T^{2} \)
47 \( 1 + (8.52 - 8.52i)T - 47iT^{2} \)
53 \( 1 + 3.01T + 53T^{2} \)
59 \( 1 + (-4.29 + 4.29i)T - 59iT^{2} \)
61 \( 1 + 14.1iT - 61T^{2} \)
67 \( 1 + (1.18 - 1.18i)T - 67iT^{2} \)
71 \( 1 + (1.96 - 1.96i)T - 71iT^{2} \)
73 \( 1 + (-8.15 + 8.15i)T - 73iT^{2} \)
79 \( 1 + 5.17T + 79T^{2} \)
83 \( 1 + (7.41 + 7.41i)T + 83iT^{2} \)
89 \( 1 + (-0.0701 + 0.0701i)T - 89iT^{2} \)
97 \( 1 + (-11.8 - 11.8i)T + 97iT^{2} \)
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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

−11.38433746338782919049810759491, −11.23893684597203898597163074949, −10.19032387591919006133902089189, −9.367830536629510639586838150329, −7.905388935396205619288315890506, −6.23208049746832183434129873663, −5.12537739716572599797964946211, −4.47132886982604743226401266367, −3.05813891608593339862832844213, −2.02168550433503671669658944819, 2.24824568770971231730452150119, 4.18680583019740767084351990732, 5.11569131618736382528760135032, 5.89471205474691614270289746215, 7.11997165201527442098365316447, 7.79018388967536634332131665505, 8.679435154220219269086908959455, 10.22740110432568203538489652974, 11.56125484358850420530532542199, 12.69743442539021832561932510838

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