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Properties

Label 2-253-253.87-c2-0-22
Degree $2$
Conductor $253$
Sign $0.643 - 0.765i$
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  + (−0.615 + 0.710i)3-s + (3.36 + 2.16i)4-s + (0.262 − 1.82i)5-s + (1.15 + 8.03i)9-s + (10.5 − 3.09i)11-s + (−3.60 + 1.05i)12-s + (1.13 + 1.30i)15-s + (6.64 + 14.5i)16-s + (4.82 − 5.57i)20-s + (−12.5 + 19.2i)23-s + (20.7 + 6.08i)25-s + (−13.5 − 8.69i)27-s + (28.7 + 33.2i)31-s + (−4.29 + 9.40i)33-s + (−13.4 + 29.5i)36-s + (−2.36 − 16.4i)37-s + ⋯
L(s)  = 1  + (−0.205 + 0.236i)3-s + (0.841 + 0.540i)4-s + (0.0524 − 0.364i)5-s + (0.128 + 0.892i)9-s + (0.959 − 0.281i)11-s + (−0.300 + 0.0882i)12-s + (0.0756 + 0.0873i)15-s + (0.415 + 0.909i)16-s + (0.241 − 0.278i)20-s + (−0.547 + 0.836i)23-s + (0.829 + 0.243i)25-s + (−0.501 − 0.322i)27-s + (0.929 + 1.07i)31-s + (−0.130 + 0.284i)33-s + (−0.374 + 0.820i)36-s + (−0.0639 − 0.444i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(6.89375\)
Root analytic conductor: \(2.62559\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :1),\ 0.643 - 0.765i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.69570 + 0.790048i\)
\(L(\frac12)\) \(\approx\) \(1.69570 + 0.790048i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-10.5 + 3.09i)T \)
23 \( 1 + (12.5 - 19.2i)T \)
good2 \( 1 + (-3.36 - 2.16i)T^{2} \)
3 \( 1 + (0.615 - 0.710i)T + (-1.28 - 8.90i)T^{2} \)
5 \( 1 + (-0.262 + 1.82i)T + (-23.9 - 7.04i)T^{2} \)
7 \( 1 + (32.0 - 37.0i)T^{2} \)
13 \( 1 + (110. + 127. i)T^{2} \)
17 \( 1 + (-120. + 262. i)T^{2} \)
19 \( 1 + (-149. - 328. i)T^{2} \)
29 \( 1 + (-349. + 765. i)T^{2} \)
31 \( 1 + (-28.7 - 33.2i)T + (-136. + 951. i)T^{2} \)
37 \( 1 + (2.36 + 16.4i)T + (-1.31e3 + 385. i)T^{2} \)
41 \( 1 + (1.61e3 + 473. i)T^{2} \)
43 \( 1 + (263. + 1.83e3i)T^{2} \)
47 \( 1 - 71.6T + 2.20e3T^{2} \)
53 \( 1 + (42.3 + 92.7i)T + (-1.83e3 + 2.12e3i)T^{2} \)
59 \( 1 + (-48.5 + 106. i)T + (-2.27e3 - 2.63e3i)T^{2} \)
61 \( 1 + (529. - 3.68e3i)T^{2} \)
67 \( 1 + (126. + 37.2i)T + (3.77e3 + 2.42e3i)T^{2} \)
71 \( 1 + (135. + 39.9i)T + (4.24e3 + 2.72e3i)T^{2} \)
73 \( 1 + (-2.21e3 - 4.84e3i)T^{2} \)
79 \( 1 + (4.08e3 + 4.71e3i)T^{2} \)
83 \( 1 + (6.60e3 - 1.94e3i)T^{2} \)
89 \( 1 + (-115. + 133. i)T + (-1.12e3 - 7.84e3i)T^{2} \)
97 \( 1 + (-25.7 + 179. i)T + (-9.02e3 - 2.65e3i)T^{2} \)
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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

−11.84123713957151881566632290730, −11.11139263090727559578810670092, −10.21055114877528576547865775102, −8.964139267502373102317281823453, −7.989293182771418773714737513786, −6.99572267936048805593015819116, −5.91090158079805622357146906748, −4.61390226861914141717539427941, −3.28666650943097015643928597924, −1.70128506675823800958693068423, 1.13105594665935182124037643053, 2.69791902219236920881109064282, 4.25472863532645696695728870452, 5.94622930451951237042750794503, 6.55413675259822337143698782971, 7.40230334578706573414250855673, 8.900148489054212052139254658729, 9.924784880537566336482698130260, 10.73637113258170331633975171190, 11.86236289528458650087528253797

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