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Properties

Label 2-221-13.10-c1-0-16
Degree $2$
Conductor $221$
Sign $-0.127 + 0.991i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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  + (1.61 − 0.929i)2-s + (−0.533 − 0.924i)3-s + (0.728 − 1.26i)4-s − 2.26i·5-s + (−1.71 − 0.992i)6-s + (−0.444 − 0.256i)7-s + 1.01i·8-s + (0.929 − 1.61i)9-s + (−2.11 − 3.65i)10-s + (−2.81 + 1.62i)11-s − 1.55·12-s + (3.58 + 0.413i)13-s − 0.953·14-s + (−2.09 + 1.21i)15-s + (2.39 + 4.14i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (1.13 − 0.657i)2-s + (−0.308 − 0.533i)3-s + (0.364 − 0.630i)4-s − 1.01i·5-s + (−0.701 − 0.405i)6-s + (−0.167 − 0.0969i)7-s + 0.357i·8-s + (0.309 − 0.536i)9-s + (−0.667 − 1.15i)10-s + (−0.849 + 0.490i)11-s − 0.449·12-s + (0.993 + 0.114i)13-s − 0.254·14-s + (−0.541 + 0.312i)15-s + (0.598 + 1.03i)16-s + (0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ -0.127 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24768 - 1.41832i\)
\(L(\frac12)\) \(\approx\) \(1.24768 - 1.41832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.58 - 0.413i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.61 + 0.929i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.533 + 0.924i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.26iT - 5T^{2} \)
7 \( 1 + (0.444 + 0.256i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.81 - 1.62i)T + (5.5 - 9.52i)T^{2} \)
19 \( 1 + (-0.234 - 0.135i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.711 - 1.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.90 - 8.50i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.02iT - 31T^{2} \)
37 \( 1 + (2.51 - 1.45i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.31 - 2.48i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.07 + 1.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.83iT - 47T^{2} \)
53 \( 1 + 8.97T + 53T^{2} \)
59 \( 1 + (-7.51 - 4.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.76 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.66 - 2.11i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.3 + 7.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.25iT - 73T^{2} \)
79 \( 1 + 4.05T + 79T^{2} \)
83 \( 1 + 1.67iT - 83T^{2} \)
89 \( 1 + (5.16 - 2.98i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.83 - 4.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.31570997990274655690089776752, −11.48340919566877931783255616705, −10.40676812875897553391412360037, −9.089365514286680790859434194933, −8.020036318095835236274304129745, −6.62276023831615646587037407210, −5.42934061809113382093771166367, −4.55265406498683146282663222606, −3.27367319490877375855126908051, −1.46406187560767731669918045794, 2.96857293952336800509760401843, 4.12739628554310815311722715334, 5.31163184850866048519388291109, 6.16095957668526321993070603698, 7.13147222858512010289751866784, 8.316615419245428271421904733208, 10.02968509799529781502252132908, 10.58945883082652887215911320472, 11.57378380057779968669160167765, 12.92321999729957679329423026732

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