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Properties

Label 2-273-13.12-c1-0-15
Degree $2$
Conductor $273$
Sign $-0.429 - 0.903i$
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  − 2.54i·2-s − 3-s − 4.49·4-s − 3.49i·5-s + 2.54i·6-s − i·7-s + 6.35i·8-s + 9-s − 8.90·10-s + 0.708i·11-s + 4.49·12-s + (−3.25 + 1.54i)13-s − 2.54·14-s + 3.49i·15-s + 7.20·16-s + 7.09·17-s + ⋯
L(s)  = 1  − 1.80i·2-s − 0.577·3-s − 2.24·4-s − 1.56i·5-s + 1.04i·6-s − 0.377i·7-s + 2.24i·8-s + 0.333·9-s − 2.81·10-s + 0.213i·11-s + 1.29·12-s + (−0.903 + 0.429i)13-s − 0.681·14-s + 0.901i·15-s + 1.80·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.429 - 0.903i$
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: \((2,\ 273,\ (\ :1/2),\ -0.429 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383405 + 0.606816i\)
\(L(\frac12)\) \(\approx\) \(0.383405 + 0.606816i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 - 1.54i)T \)
good2 \( 1 + 2.54iT - 2T^{2} \)
5 \( 1 + 3.49iT - 5T^{2} \)
11 \( 1 - 0.708iT - 11T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
19 \( 1 - 0.311iT - 19T^{2} \)
23 \( 1 + 7.88T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 7.29iT - 31T^{2} \)
37 \( 1 + 1.41iT - 37T^{2} \)
41 \( 1 + 11.8iT - 41T^{2} \)
43 \( 1 + 3.29T + 43T^{2} \)
47 \( 1 + 6.11iT - 47T^{2} \)
53 \( 1 + 3.72T + 53T^{2} \)
59 \( 1 - 2.19iT - 59T^{2} \)
61 \( 1 - 2.51T + 61T^{2} \)
67 \( 1 - 9.17iT - 67T^{2} \)
71 \( 1 + 0.708iT - 71T^{2} \)
73 \( 1 + 5.21iT - 73T^{2} \)
79 \( 1 + 2.78T + 79T^{2} \)
83 \( 1 - 6.11iT - 83T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 - 7.79iT - 97T^{2} \)
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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

−11.61165441808029078694967834137, −10.12355057560218663891081180891, −9.896426141759503732059870521470, −8.780765930652761598321662860504, −7.70087234672170235415865169373, −5.61203138308652424944243048645, −4.69181573237370921403565320532, −3.84971803648037961104394204622, −1.92744139719366819874944200990, −0.61110081856408412835972233806, 3.21293486685804373081758743109, 4.86374818263898927334881126369, 5.91801995402354912442480024432, 6.52752445078229326534894579802, 7.51249046528629412577441756403, 8.147926632089599461638946615320, 9.771107279039526941811800529864, 10.28324683594302174358427015248, 11.68633263702775756215921046041, 12.64485650077636413445991299040

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