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Properties

Label 2-241-241.225-c1-0-15
Degree $2$
Conductor $241$
Sign $0.0911 + 0.995i$
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  + (1.37 − 2.38i)2-s + (1.03 + 1.79i)3-s + (−2.79 − 4.83i)4-s + 3.43·5-s + 5.69·6-s + (−1.55 + 2.68i)7-s − 9.88·8-s + (−0.638 + 1.10i)9-s + (4.72 − 8.19i)10-s + (−1.00 + 1.74i)11-s + (5.77 − 10.0i)12-s + (−2.27 − 3.94i)13-s + (4.27 + 7.40i)14-s + (3.55 + 6.14i)15-s + (−8.02 + 13.9i)16-s − 0.322·17-s + ⋯
L(s)  = 1  + (0.973 − 1.68i)2-s + (0.597 + 1.03i)3-s + (−1.39 − 2.41i)4-s + 1.53·5-s + 2.32·6-s + (−0.586 + 1.01i)7-s − 3.49·8-s + (−0.212 + 0.368i)9-s + (1.49 − 2.59i)10-s + (−0.304 + 0.527i)11-s + (1.66 − 2.88i)12-s + (−0.631 − 1.09i)13-s + (1.14 + 1.97i)14-s + (0.916 + 1.58i)15-s + (−2.00 + 3.47i)16-s − 0.0782·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0911 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0911 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(241\)
Sign: $0.0911 + 0.995i$
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: \((2,\ 241,\ (\ :1/2),\ 0.0911 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73749 - 1.58579i\)
\(L(\frac12)\) \(\approx\) \(1.73749 - 1.58579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 + (15.4 + 1.57i)T \)
good2 \( 1 + (-1.37 + 2.38i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.03 - 1.79i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.43T + 5T^{2} \)
7 \( 1 + (1.55 - 2.68i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 - 1.74i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.27 + 3.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.322T + 17T^{2} \)
19 \( 1 + (0.852 - 1.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.90T + 23T^{2} \)
29 \( 1 + (-0.976 + 1.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.591 - 1.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.44 + 4.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.77T + 41T^{2} \)
43 \( 1 - 9.78T + 43T^{2} \)
47 \( 1 - 5.30T + 47T^{2} \)
53 \( 1 + (-5.38 - 9.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.83 + 4.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.35T + 61T^{2} \)
67 \( 1 + (0.677 - 1.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.40 + 4.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.10T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + (-2.90 + 5.02i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.411 + 0.712i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.21 + 3.82i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.24169633072139140867235643315, −10.64430439374529011848313887300, −9.961778536521288001588609649086, −9.680179278450955149772104827617, −8.792055282994074038915955898080, −5.96706720700648961372478038398, −5.42729085877591911086296997103, −4.20248534918444961083080120834, −2.85943515601623913918329633163, −2.20717851253679065469245261058, 2.49884658539891151769747754009, 4.12152332273346173244071836332, 5.50321265501570690142707991916, 6.53377092154155232245668940691, 6.97434607598704170332981612720, 7.999324199613637550696954626183, 9.022134586916339695044024450597, 10.06287263415985233348266516343, 12.08395606854602021444186989556, 13.17557591502781785174226735410

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