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Properties

Label 2-241-241.225-c1-0-0
Degree $2$
Conductor $241$
Sign $-0.0312 + 0.999i$
Analytic cond. $1.92439$
Root an. cond. $1.38722$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 + 1.96i)2-s + (0.670 + 1.16i)3-s + (−1.57 − 2.73i)4-s − 3.40·5-s − 3.04·6-s + (−0.353 + 0.612i)7-s + 2.61·8-s + (0.601 − 1.04i)9-s + (3.86 − 6.69i)10-s + (−1.78 + 3.09i)11-s + (2.11 − 3.66i)12-s + (−2.82 − 4.90i)13-s + (−0.802 − 1.39i)14-s + (−2.28 − 3.95i)15-s + (0.180 − 0.313i)16-s + 3.20·17-s + ⋯
L(s)  = 1  + (−0.802 + 1.39i)2-s + (0.386 + 0.670i)3-s + (−0.788 − 1.36i)4-s − 1.52·5-s − 1.24·6-s + (−0.133 + 0.231i)7-s + 0.925·8-s + (0.200 − 0.347i)9-s + (1.22 − 2.11i)10-s + (−0.538 + 0.933i)11-s + (0.610 − 1.05i)12-s + (−0.784 − 1.35i)13-s + (−0.214 − 0.371i)14-s + (−0.589 − 1.02i)15-s + (0.0452 − 0.0783i)16-s + 0.777·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0312 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0312 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(241\)
Sign: $-0.0312 + 0.999i$
Analytic conductor: \(1.92439\)
Root analytic conductor: \(1.38722\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{241} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 241,\ (\ :1/2),\ -0.0312 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130162 - 0.134297i\)
\(L(\frac12)\) \(\approx\) \(0.130162 - 0.134297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 + (15.1 + 3.45i)T \)
good2 \( 1 + (1.13 - 1.96i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.670 - 1.16i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.40T + 5T^{2} \)
7 \( 1 + (0.353 - 0.612i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.78 - 3.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.82 + 4.90i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.20T + 17T^{2} \)
19 \( 1 + (3.21 - 5.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.37T + 23T^{2} \)
29 \( 1 + (1.21 - 2.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.800 - 1.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.94 + 3.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.245T + 41T^{2} \)
43 \( 1 + 9.80T + 43T^{2} \)
47 \( 1 + 8.69T + 47T^{2} \)
53 \( 1 + (-1.54 - 2.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.27 - 9.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.68T + 61T^{2} \)
67 \( 1 + (2.15 - 3.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0306 + 0.0531i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.62T + 73T^{2} \)
79 \( 1 - 2.50T + 79T^{2} \)
83 \( 1 + (1.26 - 2.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.42 + 14.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.69 + 15.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.55906551887728191527815600927, −12.09286876096937717003760589905, −10.24556217198597052147636453647, −9.932283623598680132820639687034, −8.581512869976711558780123684277, −7.86815200955831648401481312018, −7.35125654488997419606447214535, −5.90429930240541975573013679395, −4.63140361917932540131752176239, −3.40336952640139002088310274064, 0.18037453013819949353352272532, 2.08557651765546303818637565977, 3.36454481678220945810769782692, 4.49203710290549466512298676866, 6.80527363186816895447480041425, 7.970685920659445118158094976098, 8.315184750471600072405774825224, 9.586853399063365927697181784932, 10.61799667641343068991020327438, 11.55258289935325284154974180892

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