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Properties

Label 2-253-11.10-c2-0-28
Degree $2$
Conductor $253$
Sign $-0.194 + 0.980i$
Analytic cond. $6.89375$
Root an. cond. $2.62559$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.34i·2-s + 3.76·3-s − 7.22·4-s + 8.66·5-s − 12.6i·6-s + 13.9i·7-s + 10.7i·8-s + 5.17·9-s − 29.0i·10-s + (2.13 − 10.7i)11-s − 27.1·12-s − 7.42i·13-s + 46.6·14-s + 32.6·15-s + 7.25·16-s + 2.62i·17-s + ⋯
L(s)  = 1  − 1.67i·2-s + 1.25·3-s − 1.80·4-s + 1.73·5-s − 2.10i·6-s + 1.98i·7-s + 1.34i·8-s + 0.575·9-s − 2.90i·10-s + (0.194 − 0.980i)11-s − 2.26·12-s − 0.571i·13-s + 3.33·14-s + 2.17·15-s + 0.453·16-s + 0.154i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.194 + 0.980i$
Analytic conductor: \(6.89375\)
Root analytic conductor: \(2.62559\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :1),\ -0.194 + 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73076 - 2.10722i\)
\(L(\frac12)\) \(\approx\) \(1.73076 - 2.10722i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-2.13 + 10.7i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 + 3.34iT - 4T^{2} \)
3 \( 1 - 3.76T + 9T^{2} \)
5 \( 1 - 8.66T + 25T^{2} \)
7 \( 1 - 13.9iT - 49T^{2} \)
13 \( 1 + 7.42iT - 169T^{2} \)
17 \( 1 - 2.62iT - 289T^{2} \)
19 \( 1 + 13.1iT - 361T^{2} \)
29 \( 1 + 1.27iT - 841T^{2} \)
31 \( 1 + 0.318T + 961T^{2} \)
37 \( 1 + 54.4T + 1.36e3T^{2} \)
41 \( 1 - 64.1iT - 1.68e3T^{2} \)
43 \( 1 + 39.4iT - 1.84e3T^{2} \)
47 \( 1 - 33.0T + 2.20e3T^{2} \)
53 \( 1 + 14.4T + 2.80e3T^{2} \)
59 \( 1 + 68.1T + 3.48e3T^{2} \)
61 \( 1 - 88.2iT - 3.72e3T^{2} \)
67 \( 1 + 18.4T + 4.48e3T^{2} \)
71 \( 1 - 107.T + 5.04e3T^{2} \)
73 \( 1 - 45.0iT - 5.32e3T^{2} \)
79 \( 1 + 66.7iT - 6.24e3T^{2} \)
83 \( 1 + 69.3iT - 6.88e3T^{2} \)
89 \( 1 - 17.7T + 7.92e3T^{2} \)
97 \( 1 - 95.3T + 9.40e3T^{2} \)
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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

−11.58278236467132890494914416159, −10.46360594753552449963731610238, −9.514444171706119063724891931774, −8.974511790723379099955066613256, −8.495143854696781444009907735691, −6.11036129225440806511602483662, −5.20578388667508787223289004257, −3.16881969903372874140545221319, −2.58479676717091335274572791516, −1.75878609194942520765258831341, 1.83644257859909601327659218289, 3.86161468077643555572512680822, 5.01639714279037592707042708870, 6.40005962987258757210842974028, 7.11951467497828429089829933778, 7.914242414217063619101203727218, 9.116050112150733258326044137837, 9.696127615276244933273198808677, 10.53447835327871493853708647823, 12.82193641812283217616146108213

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