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Properties

Label 2-253-11.10-c2-0-31
Degree $2$
Conductor $253$
Sign $0.450 + 0.892i$
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  − 1.61i·2-s + 2.50·3-s + 1.38·4-s + 5.13·5-s − 4.05i·6-s + 0.928i·7-s − 8.70i·8-s − 2.72·9-s − 8.30i·10-s + (4.95 + 9.82i)11-s + 3.46·12-s − 16.1i·13-s + 1.50·14-s + 12.8·15-s − 8.56·16-s + 11.2i·17-s + ⋯
L(s)  = 1  − 0.808i·2-s + 0.835·3-s + 0.345·4-s + 1.02·5-s − 0.675i·6-s + 0.132i·7-s − 1.08i·8-s − 0.302·9-s − 0.830i·10-s + (0.450 + 0.892i)11-s + 0.288·12-s − 1.23i·13-s + 0.107·14-s + 0.857·15-s − 0.535·16-s + 0.664i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.450 + 0.892i$
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: \((2,\ 253,\ (\ :1),\ 0.450 + 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.27680 - 1.40210i\)
\(L(\frac12)\) \(\approx\) \(2.27680 - 1.40210i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-4.95 - 9.82i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 + 1.61iT - 4T^{2} \)
3 \( 1 - 2.50T + 9T^{2} \)
5 \( 1 - 5.13T + 25T^{2} \)
7 \( 1 - 0.928iT - 49T^{2} \)
13 \( 1 + 16.1iT - 169T^{2} \)
17 \( 1 - 11.2iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
29 \( 1 - 3.71iT - 841T^{2} \)
31 \( 1 - 43.1T + 961T^{2} \)
37 \( 1 + 63.2T + 1.36e3T^{2} \)
41 \( 1 + 67.1iT - 1.68e3T^{2} \)
43 \( 1 - 24.9iT - 1.84e3T^{2} \)
47 \( 1 + 80.7T + 2.20e3T^{2} \)
53 \( 1 + 68.1T + 2.80e3T^{2} \)
59 \( 1 - 55.9T + 3.48e3T^{2} \)
61 \( 1 - 86.8iT - 3.72e3T^{2} \)
67 \( 1 - 76.7T + 4.48e3T^{2} \)
71 \( 1 + 57.8T + 5.04e3T^{2} \)
73 \( 1 - 135. iT - 5.32e3T^{2} \)
79 \( 1 + 53.3iT - 6.24e3T^{2} \)
83 \( 1 + 29.6iT - 6.88e3T^{2} \)
89 \( 1 - 126.T + 7.92e3T^{2} \)
97 \( 1 + 1.53T + 9.40e3T^{2} \)
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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

−11.72656850759625871497643449510, −10.38409737915274024115824073971, −10.00819250167606751559392486329, −8.953600497211117807910888563866, −7.87569724352061832306069875185, −6.59463877492961183130569618363, −5.51382538948332772697243881147, −3.70520646353763959890232301188, −2.61416480565156051589702769591, −1.63768124299359300200737060304, 1.95326759964789872310615644458, 3.12092561302020622440373827583, 4.98033178183812916884304696738, 6.19098003519025689326582859919, 6.83234982886706257415104358751, 8.135480776312594881541755876178, 8.918726264366612545025043260369, 9.719148100525828325753445347263, 11.13975382193851598003655213433, 11.79683786019835626928267243983

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