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Properties

Label 2-253-253.54-c2-0-7
Degree $2$
Conductor $253$
Sign $0.890 - 0.455i$
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  + (−2.49 − 5.45i)3-s + (−3.83 − 1.12i)4-s + (6.35 + 7.33i)5-s + (−17.6 + 20.3i)9-s + (1.56 + 10.8i)11-s + (3.41 + 23.7i)12-s + (24.1 − 52.9i)15-s + (13.4 + 8.65i)16-s + (−16.1 − 35.3i)20-s + (12.2 − 19.4i)23-s + (−9.84 + 68.5i)25-s + (103. + 30.4i)27-s + (−5.55 + 12.1i)31-s + (55.5 − 35.6i)33-s + (90.8 − 58.3i)36-s + (−28.5 + 32.9i)37-s + ⋯
L(s)  = 1  + (−0.830 − 1.81i)3-s + (−0.959 − 0.281i)4-s + (1.27 + 1.46i)5-s + (−1.96 + 2.26i)9-s + (0.142 + 0.989i)11-s + (0.284 + 1.97i)12-s + (1.61 − 3.53i)15-s + (0.841 + 0.540i)16-s + (−0.806 − 1.76i)20-s + (0.534 − 0.845i)23-s + (−0.393 + 2.74i)25-s + (3.83 + 1.12i)27-s + (−0.179 + 0.392i)31-s + (1.68 − 1.08i)33-s + (2.52 − 1.62i)36-s + (−0.771 + 0.890i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(6.89375\)
Root analytic conductor: \(2.62559\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 253,\ (\ :1),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.889330 + 0.214351i\)
\(L(\frac12)\) \(\approx\) \(0.889330 + 0.214351i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-1.56 - 10.8i)T \)
23 \( 1 + (-12.2 + 19.4i)T \)
good2 \( 1 + (3.83 + 1.12i)T^{2} \)
3 \( 1 + (2.49 + 5.45i)T + (-5.89 + 6.80i)T^{2} \)
5 \( 1 + (-6.35 - 7.33i)T + (-3.55 + 24.7i)T^{2} \)
7 \( 1 + (-20.3 - 44.5i)T^{2} \)
13 \( 1 + (-70.2 + 153. i)T^{2} \)
17 \( 1 + (-243. + 156. i)T^{2} \)
19 \( 1 + (-303. - 195. i)T^{2} \)
29 \( 1 + (-707. + 454. i)T^{2} \)
31 \( 1 + (5.55 - 12.1i)T + (-629. - 726. i)T^{2} \)
37 \( 1 + (28.5 - 32.9i)T + (-194. - 1.35e3i)T^{2} \)
41 \( 1 + (239. - 1.66e3i)T^{2} \)
43 \( 1 + (1.21e3 - 1.39e3i)T^{2} \)
47 \( 1 - 27.4T + 2.20e3T^{2} \)
53 \( 1 + (-37.6 - 24.1i)T + (1.16e3 + 2.55e3i)T^{2} \)
59 \( 1 + (74.5 - 47.9i)T + (1.44e3 - 3.16e3i)T^{2} \)
61 \( 1 + (2.43e3 + 2.81e3i)T^{2} \)
67 \( 1 + (-5.76 + 40.0i)T + (-4.30e3 - 1.26e3i)T^{2} \)
71 \( 1 + (9.70 - 67.4i)T + (-4.83e3 - 1.42e3i)T^{2} \)
73 \( 1 + (-4.48e3 - 2.88e3i)T^{2} \)
79 \( 1 + (-2.59e3 + 5.67e3i)T^{2} \)
83 \( 1 + (980. + 6.81e3i)T^{2} \)
89 \( 1 + (0.378 + 0.829i)T + (-5.18e3 + 5.98e3i)T^{2} \)
97 \( 1 + (-124. - 143. i)T + (-1.33e3 + 9.31e3i)T^{2} \)
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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

−12.09609182372278753980835649125, −10.83104113729997099749086627264, −10.26260674647533800796335668481, −8.974266715212764216582477185399, −7.55714318725869813465983475055, −6.75337229530578919642560152938, −6.04332305333306841792924048442, −5.09293235053050385985955112254, −2.63247617353325735809068674864, −1.46410750649385793680854809922, 0.58104167670097708620732072976, 3.56959985545802336590316908732, 4.62968317598652582468627007556, 5.38943834432353978658108001967, 5.92173158585242542588529248374, 8.579881530381908167951327841805, 9.059906387921310094858419963216, 9.681018079801420220752082694742, 10.50123393497857305232614441627, 11.68191991618183730473953099905

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