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Properties

Label 2-253-11.10-c2-0-14
Degree $2$
Conductor $253$
Sign $0.887 - 0.461i$
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  + 3.51i·2-s − 4.54·3-s − 8.38·4-s − 6.03·5-s − 15.9i·6-s + 3.84i·7-s − 15.4i·8-s + 11.6·9-s − 21.2i·10-s + (−9.75 + 5.07i)11-s + 38.0·12-s − 10.2i·13-s − 13.5·14-s + 27.4·15-s + 20.7·16-s + 14.5i·17-s + ⋯
L(s)  = 1  + 1.75i·2-s − 1.51·3-s − 2.09·4-s − 1.20·5-s − 2.66i·6-s + 0.549i·7-s − 1.92i·8-s + 1.29·9-s − 2.12i·10-s + (−0.887 + 0.461i)11-s + 3.17·12-s − 0.787i·13-s − 0.966·14-s + 1.82·15-s + 1.29·16-s + 0.858i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.887 - 0.461i$
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: \((2,\ 253,\ (\ :1),\ 0.887 - 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.124644 + 0.0304942i\)
\(L(\frac12)\) \(\approx\) \(0.124644 + 0.0304942i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (9.75 - 5.07i)T \)
23 \( 1 + 4.79T \)
good2 \( 1 - 3.51iT - 4T^{2} \)
3 \( 1 + 4.54T + 9T^{2} \)
5 \( 1 + 6.03T + 25T^{2} \)
7 \( 1 - 3.84iT - 49T^{2} \)
13 \( 1 + 10.2iT - 169T^{2} \)
17 \( 1 - 14.5iT - 289T^{2} \)
19 \( 1 - 25.7iT - 361T^{2} \)
29 \( 1 + 3.68iT - 841T^{2} \)
31 \( 1 - 47.7T + 961T^{2} \)
37 \( 1 + 22.3T + 1.36e3T^{2} \)
41 \( 1 + 42.7iT - 1.68e3T^{2} \)
43 \( 1 + 52.1iT - 1.84e3T^{2} \)
47 \( 1 + 4.97T + 2.20e3T^{2} \)
53 \( 1 + 50.4T + 2.80e3T^{2} \)
59 \( 1 + 98.9T + 3.48e3T^{2} \)
61 \( 1 + 64.8iT - 3.72e3T^{2} \)
67 \( 1 - 68.1T + 4.48e3T^{2} \)
71 \( 1 - 83.2T + 5.04e3T^{2} \)
73 \( 1 - 128. iT - 5.32e3T^{2} \)
79 \( 1 + 65.3iT - 6.24e3T^{2} \)
83 \( 1 - 136. iT - 6.88e3T^{2} \)
89 \( 1 - 40.2T + 7.92e3T^{2} \)
97 \( 1 + 102.T + 9.40e3T^{2} \)
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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

−12.26020130920342330381001470284, −10.91235172792010710880286445934, −9.993063759860860315715488344771, −8.316404301125427980622634932905, −7.85369255319562823110205857578, −6.75767367413152110166500934554, −5.79056696730006407425632430346, −5.16130944840082976299343726108, −4.03233396672314823097012288614, −0.12348871081222184754865710619, 0.831524721330992173963576158128, 2.96606539488841695077982730765, 4.38373308914269352511337170870, 4.92845332922931249301328093477, 6.65007950276306985350402846012, 7.928449557200686496717999261040, 9.306329612207109949814372828288, 10.42591586454930181852166127014, 11.07862228554133224956589183340, 11.60283050408661172989025486043

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