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Properties

Label 2-221-13.12-c1-0-10
Degree $2$
Conductor $221$
Sign $0.792 + 0.609i$
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  − 1.26i·2-s + 1.60·3-s + 0.411·4-s + 3.67i·5-s − 2.02i·6-s − 1.74i·7-s − 3.03i·8-s − 0.411·9-s + 4.62·10-s + 1.01i·11-s + 0.661·12-s + (2.85 + 2.19i)13-s − 2.19·14-s + 5.90i·15-s − 3.00·16-s + 17-s + ⋯
L(s)  = 1  − 0.891i·2-s + 0.928·3-s + 0.205·4-s + 1.64i·5-s − 0.827i·6-s − 0.659i·7-s − 1.07i·8-s − 0.137·9-s + 1.46·10-s + 0.306i·11-s + 0.191·12-s + (0.792 + 0.609i)13-s − 0.587·14-s + 1.52i·15-s − 0.752·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63221 - 0.555454i\)
\(L(\frac12)\) \(\approx\) \(1.63221 - 0.555454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.85 - 2.19i)T \)
17 \( 1 - T \)
good2 \( 1 + 1.26iT - 2T^{2} \)
3 \( 1 - 1.60T + 3T^{2} \)
5 \( 1 - 3.67iT - 5T^{2} \)
7 \( 1 + 1.74iT - 7T^{2} \)
11 \( 1 - 1.01iT - 11T^{2} \)
19 \( 1 + 7.06iT - 19T^{2} \)
23 \( 1 + 8.79T + 23T^{2} \)
29 \( 1 - 2.10T + 29T^{2} \)
31 \( 1 + 1.40iT - 31T^{2} \)
37 \( 1 - 3.67iT - 37T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 + 3.91iT - 47T^{2} \)
53 \( 1 - 9.18T + 53T^{2} \)
59 \( 1 + 2.70iT - 59T^{2} \)
61 \( 1 - 7.22T + 61T^{2} \)
67 \( 1 + 3.34iT - 67T^{2} \)
71 \( 1 - 8.50iT - 71T^{2} \)
73 \( 1 + 7.15iT - 73T^{2} \)
79 \( 1 - 7.51T + 79T^{2} \)
83 \( 1 - 4.63iT - 83T^{2} \)
89 \( 1 - 10.8iT - 89T^{2} \)
97 \( 1 + 5.36iT - 97T^{2} \)
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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

−11.74492350438677865289810191876, −11.25736749592012487535908468119, −10.30541085223735804599129435079, −9.650280766177241907086074045976, −8.204875239092516182903110859286, −7.06574963209846354709031010634, −6.40514834458712987542583470145, −3.97026372980472442228834927424, −3.10094042964455287222840793490, −2.14768034129501570981883970795, 1.97073639985687418739234956017, 3.74499417523636101805344812309, 5.46669570483448309297171937464, 5.93063871581745596280975317227, 7.80447219160625809185023847793, 8.413372090441467035272689032781, 8.814731320400127590363145633387, 10.16754133347510684172093154039, 11.77129546186515181722902660995, 12.38375673918883250992474266812

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