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Properties

Label 2-253-11.10-c2-0-33
Degree $2$
Conductor $253$
Sign $0.831 + 0.555i$
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  − 0.919i·2-s + 4.75·3-s + 3.15·4-s + 0.241·5-s − 4.37i·6-s + 3.10i·7-s − 6.58i·8-s + 13.5·9-s − 0.222i·10-s + (−9.14 − 6.11i)11-s + 14.9·12-s + 4.95i·13-s + 2.85·14-s + 1.14·15-s + 6.56·16-s − 23.5i·17-s + ⋯
L(s)  = 1  − 0.459i·2-s + 1.58·3-s + 0.788·4-s + 0.0482·5-s − 0.728i·6-s + 0.442i·7-s − 0.822i·8-s + 1.51·9-s − 0.0222i·10-s + (−0.831 − 0.555i)11-s + 1.24·12-s + 0.381i·13-s + 0.203·14-s + 0.0764·15-s + 0.410·16-s − 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.831 + 0.555i$
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: \((2,\ 253,\ (\ :1),\ 0.831 + 0.555i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.89515 - 0.878949i\)
\(L(\frac12)\) \(\approx\) \(2.89515 - 0.878949i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (9.14 + 6.11i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 + 0.919iT - 4T^{2} \)
3 \( 1 - 4.75T + 9T^{2} \)
5 \( 1 - 0.241T + 25T^{2} \)
7 \( 1 - 3.10iT - 49T^{2} \)
13 \( 1 - 4.95iT - 169T^{2} \)
17 \( 1 + 23.5iT - 289T^{2} \)
19 \( 1 - 32.7iT - 361T^{2} \)
29 \( 1 - 47.1iT - 841T^{2} \)
31 \( 1 - 25.6T + 961T^{2} \)
37 \( 1 + 50.3T + 1.36e3T^{2} \)
41 \( 1 + 13.0iT - 1.68e3T^{2} \)
43 \( 1 + 48.3iT - 1.84e3T^{2} \)
47 \( 1 - 30.1T + 2.20e3T^{2} \)
53 \( 1 - 5.53T + 2.80e3T^{2} \)
59 \( 1 + 32.5T + 3.48e3T^{2} \)
61 \( 1 + 83.2iT - 3.72e3T^{2} \)
67 \( 1 + 127.T + 4.48e3T^{2} \)
71 \( 1 - 40.3T + 5.04e3T^{2} \)
73 \( 1 - 57.4iT - 5.32e3T^{2} \)
79 \( 1 - 76.1iT - 6.24e3T^{2} \)
83 \( 1 - 115. iT - 6.88e3T^{2} \)
89 \( 1 + 46.9T + 7.92e3T^{2} \)
97 \( 1 + 14.0T + 9.40e3T^{2} \)
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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

−11.87301785884365088882484604140, −10.60640669257089214097638060410, −9.809514222917220888643040508072, −8.815165906130787124844207509896, −7.906655268231652601396316851394, −7.04911086322654761955146013953, −5.57184635051785182892599675976, −3.70579577909634079438928489912, −2.83138906507665687286175078273, −1.83555093570280436255096303003, 2.03120355382228717685456172265, 2.97827659073528614468808206168, 4.40254305238262775673651999358, 6.03816831448492554399296574476, 7.30974081044183680662355442856, 7.86697355310214587785178507125, 8.708875117910097327353508478672, 9.950363325042522800774025239588, 10.69392785495282160385174966683, 11.99797043027612119785203755669

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