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Properties

Label 2-273-39.8-c1-0-19
Degree $2$
Conductor $273$
Sign $0.601 + 0.798i$
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  + (0.217 − 0.217i)2-s + (0.368 − 1.69i)3-s + 1.90i·4-s + (1.58 − 1.58i)5-s + (−0.287 − 0.447i)6-s + (0.707 − 0.707i)7-s + (0.848 + 0.848i)8-s + (−2.72 − 1.24i)9-s − 0.690i·10-s + (0.523 + 0.523i)11-s + (3.22 + 0.702i)12-s + (3.36 − 1.29i)13-s − 0.307i·14-s + (−2.10 − 3.27i)15-s − 3.44·16-s − 0.407·17-s + ⋯
L(s)  = 1  + (0.153 − 0.153i)2-s + (0.212 − 0.977i)3-s + 0.952i·4-s + (0.710 − 0.710i)5-s + (−0.117 − 0.182i)6-s + (0.267 − 0.267i)7-s + (0.300 + 0.300i)8-s + (−0.909 − 0.415i)9-s − 0.218i·10-s + (0.157 + 0.157i)11-s + (0.930 + 0.202i)12-s + (0.932 − 0.360i)13-s − 0.0821i·14-s + (−0.542 − 0.845i)15-s − 0.860·16-s − 0.0989·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.601 + 0.798i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.601 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46974 - 0.733188i\)
\(L(\frac12)\) \(\approx\) \(1.46974 - 0.733188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.368 + 1.69i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-3.36 + 1.29i)T \)
good2 \( 1 + (-0.217 + 0.217i)T - 2iT^{2} \)
5 \( 1 + (-1.58 + 1.58i)T - 5iT^{2} \)
11 \( 1 + (-0.523 - 0.523i)T + 11iT^{2} \)
17 \( 1 + 0.407T + 17T^{2} \)
19 \( 1 + (0.544 + 0.544i)T + 19iT^{2} \)
23 \( 1 + 1.21T + 23T^{2} \)
29 \( 1 + 9.47iT - 29T^{2} \)
31 \( 1 + (-3.76 - 3.76i)T + 31iT^{2} \)
37 \( 1 + (2.47 - 2.47i)T - 37iT^{2} \)
41 \( 1 + (5.01 - 5.01i)T - 41iT^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 + (3.18 + 3.18i)T + 47iT^{2} \)
53 \( 1 - 8.86iT - 53T^{2} \)
59 \( 1 + (-5.08 - 5.08i)T + 59iT^{2} \)
61 \( 1 + 1.99T + 61T^{2} \)
67 \( 1 + (-1.89 - 1.89i)T + 67iT^{2} \)
71 \( 1 + (6.34 - 6.34i)T - 71iT^{2} \)
73 \( 1 + (-3.00 + 3.00i)T - 73iT^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 + (11.7 - 11.7i)T - 83iT^{2} \)
89 \( 1 + (9.46 + 9.46i)T + 89iT^{2} \)
97 \( 1 + (-1.44 - 1.44i)T + 97iT^{2} \)
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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

−11.94088013404208205838091111958, −11.18298077331690364005502015176, −9.702896279214725724595040045406, −8.542565420515879319235728978701, −8.083020493336227721958120066437, −6.88957197852644128270672324972, −5.82664625876807534888022997699, −4.39680304593791076633891830432, −2.93997527571336756821270539978, −1.50308702044944089781921600537, 2.06590138325509989562073795277, 3.66418530793899826516162626938, 5.01975117302220226379110007467, 5.89819556976023981125033919408, 6.77992137392047972599258883980, 8.522540604514219149663804426968, 9.313060005858234294952644427441, 10.31809693883784008704813041724, 10.74190114394888591646168478577, 11.70449211628529545241644484769

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