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Properties

Label 2-253-11.10-c2-0-16
Degree $2$
Conductor $253$
Sign $-0.699 - 0.714i$
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  + 2.50i·2-s + 5.31·3-s − 2.28·4-s − 8.14·5-s + 13.3i·6-s + 2.32i·7-s + 4.30i·8-s + 19.2·9-s − 20.4i·10-s + (7.69 + 7.86i)11-s − 12.1·12-s + 19.3i·13-s − 5.81·14-s − 43.2·15-s − 19.9·16-s − 26.1i·17-s + ⋯
L(s)  = 1  + 1.25i·2-s + 1.77·3-s − 0.570·4-s − 1.62·5-s + 2.22i·6-s + 0.331i·7-s + 0.538i·8-s + 2.13·9-s − 2.04i·10-s + (0.699 + 0.714i)11-s − 1.01·12-s + 1.48i·13-s − 0.415·14-s − 2.88·15-s − 1.24·16-s − 1.53i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.699 - 0.714i$
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: \((2,\ 253,\ (\ :1),\ -0.699 - 0.714i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.890319 + 2.11688i\)
\(L(\frac12)\) \(\approx\) \(0.890319 + 2.11688i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-7.69 - 7.86i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 - 2.50iT - 4T^{2} \)
3 \( 1 - 5.31T + 9T^{2} \)
5 \( 1 + 8.14T + 25T^{2} \)
7 \( 1 - 2.32iT - 49T^{2} \)
13 \( 1 - 19.3iT - 169T^{2} \)
17 \( 1 + 26.1iT - 289T^{2} \)
19 \( 1 - 7.39iT - 361T^{2} \)
29 \( 1 + 21.0iT - 841T^{2} \)
31 \( 1 - 30.2T + 961T^{2} \)
37 \( 1 - 13.8T + 1.36e3T^{2} \)
41 \( 1 + 30.0iT - 1.68e3T^{2} \)
43 \( 1 + 46.1iT - 1.84e3T^{2} \)
47 \( 1 + 48.3T + 2.20e3T^{2} \)
53 \( 1 + 1.13T + 2.80e3T^{2} \)
59 \( 1 - 39.8T + 3.48e3T^{2} \)
61 \( 1 - 1.89iT - 3.72e3T^{2} \)
67 \( 1 - 102.T + 4.48e3T^{2} \)
71 \( 1 - 49.8T + 5.04e3T^{2} \)
73 \( 1 - 59.8iT - 5.32e3T^{2} \)
79 \( 1 + 88.4iT - 6.24e3T^{2} \)
83 \( 1 + 135. iT - 6.88e3T^{2} \)
89 \( 1 + 144.T + 7.92e3T^{2} \)
97 \( 1 + 70.9T + 9.40e3T^{2} \)
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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

−12.09134497795918491328802362858, −11.55530330369234034205248064327, −9.627282933948991772427892021429, −8.850058972109919690911569662437, −8.141510359458285501444807780191, −7.30415432636394492365815984329, −6.81965339818712239436725628499, −4.64532142321055983614402079052, −3.85770645624123102001027646231, −2.34341021226109657638981022335, 1.08764924655233489092839134305, 2.87722652645863949132567241150, 3.57094998948290198793304565761, 4.21257678042299103521895625796, 6.83637052941288898328376102951, 8.125857272042977522986899235993, 8.316805899768466941189248949048, 9.623886981881320243453453593398, 10.59741427894985310870383242108, 11.37415723534373899109338090153

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