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Properties

Label 2-273-91.31-c1-0-14
Degree $2$
Conductor $273$
Sign $0.542 + 0.840i$
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  + (0.612 + 0.164i)2-s + (0.866 + 0.5i)3-s + (−1.38 − 0.798i)4-s + (0.690 − 2.57i)5-s + (0.448 + 0.448i)6-s + (−1.89 − 1.84i)7-s + (−1.61 − 1.61i)8-s + (0.499 + 0.866i)9-s + (0.845 − 1.46i)10-s + (4.30 − 1.15i)11-s + (−0.798 − 1.38i)12-s + (2.17 − 2.87i)13-s + (−0.860 − 1.44i)14-s + (1.88 − 1.88i)15-s + (0.874 + 1.51i)16-s + (−2.00 + 3.47i)17-s + ⋯
L(s)  = 1  + (0.433 + 0.116i)2-s + (0.499 + 0.288i)3-s + (−0.691 − 0.399i)4-s + (0.308 − 1.15i)5-s + (0.183 + 0.183i)6-s + (−0.717 − 0.696i)7-s + (−0.570 − 0.570i)8-s + (0.166 + 0.288i)9-s + (0.267 − 0.462i)10-s + (1.29 − 0.347i)11-s + (−0.230 − 0.399i)12-s + (0.603 − 0.797i)13-s + (−0.229 − 0.384i)14-s + (0.486 − 0.486i)15-s + (0.218 + 0.378i)16-s + (−0.486 + 0.843i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.542 + 0.840i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.542 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36133 - 0.741616i\)
\(L(\frac12)\) \(\approx\) \(1.36133 - 0.741616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.89 + 1.84i)T \)
13 \( 1 + (-2.17 + 2.87i)T \)
good2 \( 1 + (-0.612 - 0.164i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.690 + 2.57i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.30 + 1.15i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.00 - 3.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.13 - 4.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.04 - 1.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.379T + 29T^{2} \)
31 \( 1 + (-10.1 + 2.71i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.0661 - 0.246i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.78 + 7.78i)T + 41iT^{2} \)
43 \( 1 + 3.61iT - 43T^{2} \)
47 \( 1 + (-4.20 - 1.12i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.54 - 9.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.924 - 3.44i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-12.0 + 6.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.38 - 12.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.43 - 5.43i)T - 71iT^{2} \)
73 \( 1 + (2.09 + 7.80i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.48 - 2.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.78 - 8.78i)T + 83iT^{2} \)
89 \( 1 + (0.621 + 0.166i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.39 + 7.39i)T + 97iT^{2} \)
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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

−12.11761660789177925818071977646, −10.47050646110790439163464137956, −9.783814093382931746709421810796, −8.880108702647391257122912966072, −8.259137177722514214515854829833, −6.45870791879356054575872217341, −5.63182862973319928729527386548, −4.24533025631660756209020055480, −3.68850596665560178539827838332, −1.14352570430804101671338640961, 2.45763955536716423184938119899, 3.40808393196723065753885092504, 4.61808254697820541460974749709, 6.43425695343965082357737204692, 6.73716329870380588937585285355, 8.408431560788262086187338698293, 9.188859906568993096151938416889, 9.907933247835909026474903501304, 11.47928454868064229325501928629, 12.02706317234884312975755367322

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