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Properties

Label 2-253-11.10-c2-0-29
Degree $2$
Conductor $253$
Sign $0.808 + 0.588i$
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  + 1.81i·2-s − 2.41·3-s + 0.713·4-s + 4.05·5-s − 4.37i·6-s − 13.5i·7-s + 8.54i·8-s − 3.17·9-s + 7.34i·10-s + (−8.89 − 6.47i)11-s − 1.72·12-s − 22.1i·13-s + 24.4·14-s − 9.78·15-s − 12.6·16-s + 2.51i·17-s + ⋯
L(s)  = 1  + 0.906i·2-s − 0.804·3-s + 0.178·4-s + 0.810·5-s − 0.729i·6-s − 1.92i·7-s + 1.06i·8-s − 0.352·9-s + 0.734i·10-s + (−0.808 − 0.588i)11-s − 0.143·12-s − 1.70i·13-s + 1.74·14-s − 0.652·15-s − 0.789·16-s + 0.148i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.808 + 0.588i$
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: \((2,\ 253,\ (\ :1),\ 0.808 + 0.588i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.15187 - 0.374805i\)
\(L(\frac12)\) \(\approx\) \(1.15187 - 0.374805i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (8.89 + 6.47i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 - 1.81iT - 4T^{2} \)
3 \( 1 + 2.41T + 9T^{2} \)
5 \( 1 - 4.05T + 25T^{2} \)
7 \( 1 + 13.5iT - 49T^{2} \)
13 \( 1 + 22.1iT - 169T^{2} \)
17 \( 1 - 2.51iT - 289T^{2} \)
19 \( 1 + 18.9iT - 361T^{2} \)
29 \( 1 - 44.4iT - 841T^{2} \)
31 \( 1 - 58.9T + 961T^{2} \)
37 \( 1 - 27.8T + 1.36e3T^{2} \)
41 \( 1 + 35.5iT - 1.68e3T^{2} \)
43 \( 1 + 23.8iT - 1.84e3T^{2} \)
47 \( 1 + 18.3T + 2.20e3T^{2} \)
53 \( 1 + 10.6T + 2.80e3T^{2} \)
59 \( 1 - 12.3T + 3.48e3T^{2} \)
61 \( 1 - 33.3iT - 3.72e3T^{2} \)
67 \( 1 - 97.8T + 4.48e3T^{2} \)
71 \( 1 + 92.8T + 5.04e3T^{2} \)
73 \( 1 - 59.0iT - 5.32e3T^{2} \)
79 \( 1 + 82.0iT - 6.24e3T^{2} \)
83 \( 1 - 26.5iT - 6.88e3T^{2} \)
89 \( 1 - 28.0T + 7.92e3T^{2} \)
97 \( 1 - 54.0T + 9.40e3T^{2} \)
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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

−11.42822259386098573082594804948, −10.59842350051955697717219236403, −10.25868143392952325126766686115, −8.416312285675155369432580693553, −7.56123355774180777346553048057, −6.61041044618905311783284873914, −5.72389038151155898348591884794, −4.92952150009889541707984818932, −2.96613948138804081514354623760, −0.66297517991029214456001012008, 1.93971096835823087078090457115, 2.64647919345810754359469459312, 4.68927096150755117192681309527, 5.97753023333632246730555972250, 6.36052728904400159196611695929, 8.199477937702869170533707662943, 9.521229274946354392296020159221, 9.941117692062691845218146661341, 11.33648444666270011088424814178, 11.81220817636649018582114065116

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