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Properties

Label 2-253-11.10-c2-0-7
Degree $2$
Conductor $253$
Sign $-0.856 + 0.516i$
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  + 3.08i·2-s − 3.11·3-s − 5.53·4-s + 8.57·5-s − 9.62i·6-s + 4.55i·7-s − 4.72i·8-s + 0.725·9-s + 26.4i·10-s + (−9.41 + 5.68i)11-s + 17.2·12-s + 5.40i·13-s − 14.0·14-s − 26.7·15-s − 7.52·16-s + 30.3i·17-s + ⋯
L(s)  = 1  + 1.54i·2-s − 1.03·3-s − 1.38·4-s + 1.71·5-s − 1.60i·6-s + 0.650i·7-s − 0.591i·8-s + 0.0805·9-s + 2.64i·10-s + (−0.856 + 0.516i)11-s + 1.43·12-s + 0.415i·13-s − 1.00·14-s − 1.78·15-s − 0.470·16-s + 1.78i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $-0.856 + 0.516i$
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: \((2,\ 253,\ (\ :1),\ -0.856 + 0.516i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.269671 - 0.968965i\)
\(L(\frac12)\) \(\approx\) \(0.269671 - 0.968965i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (9.41 - 5.68i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 - 3.08iT - 4T^{2} \)
3 \( 1 + 3.11T + 9T^{2} \)
5 \( 1 - 8.57T + 25T^{2} \)
7 \( 1 - 4.55iT - 49T^{2} \)
13 \( 1 - 5.40iT - 169T^{2} \)
17 \( 1 - 30.3iT - 289T^{2} \)
19 \( 1 + 32.2iT - 361T^{2} \)
29 \( 1 + 3.47iT - 841T^{2} \)
31 \( 1 + 51.2T + 961T^{2} \)
37 \( 1 + 47.7T + 1.36e3T^{2} \)
41 \( 1 - 68.9iT - 1.68e3T^{2} \)
43 \( 1 + 3.80iT - 1.84e3T^{2} \)
47 \( 1 + 55.3T + 2.20e3T^{2} \)
53 \( 1 - 45.7T + 2.80e3T^{2} \)
59 \( 1 - 43.9T + 3.48e3T^{2} \)
61 \( 1 + 17.4iT - 3.72e3T^{2} \)
67 \( 1 - 78.2T + 4.48e3T^{2} \)
71 \( 1 - 15.7T + 5.04e3T^{2} \)
73 \( 1 - 66.2iT - 5.32e3T^{2} \)
79 \( 1 - 66.9iT - 6.24e3T^{2} \)
83 \( 1 + 38.9iT - 6.88e3T^{2} \)
89 \( 1 - 70.6T + 7.92e3T^{2} \)
97 \( 1 - 18.9T + 9.40e3T^{2} \)
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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

−12.78942184027934712763236185441, −11.28700154645245989400242354966, −10.35484263417313479667147408652, −9.252706948469484873204621636431, −8.469084332820306287234311479429, −6.92590664920782996150993463584, −6.26339723422382665522198279345, −5.46676864245316316617613341406, −4.97905292984170535141348905873, −2.17960155854401817524238328853, 0.57082405665431678478609623005, 2.00768603141348098634055050725, 3.30777956580342262207839596300, 5.13970465782908961028460223738, 5.68060529945723533621318722615, 7.05848444205688183104029123733, 8.875060055284271929969341219954, 9.931270395932595027697752513269, 10.44194577106220172006192388899, 11.04930134974160605068578260490

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