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Properties

Label 2-221-13.4-c1-0-14
Degree $2$
Conductor $221$
Sign $0.933 - 0.358i$
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  + (1.98 + 1.14i)2-s + (1.31 − 2.26i)3-s + (1.62 + 2.81i)4-s + 1.31i·5-s + (5.20 − 3.00i)6-s + (−3.37 + 1.94i)7-s + 2.87i·8-s + (−1.93 − 3.35i)9-s + (−1.50 + 2.61i)10-s + (−0.635 − 0.366i)11-s + 8.52·12-s + (−1.24 − 3.38i)13-s − 8.93·14-s + (2.98 + 1.72i)15-s + (−0.0405 + 0.0702i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (1.40 + 0.810i)2-s + (0.756 − 1.31i)3-s + (0.813 + 1.40i)4-s + 0.588i·5-s + (2.12 − 1.22i)6-s + (−1.27 + 0.736i)7-s + 1.01i·8-s + (−0.644 − 1.11i)9-s + (−0.477 + 0.826i)10-s + (−0.191 − 0.110i)11-s + 2.46·12-s + (−0.346 − 0.938i)13-s − 2.38·14-s + (0.771 + 0.445i)15-s + (−0.0101 + 0.0175i)16-s + (−0.121 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 221,\ (\ :1/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61177 + 0.484438i\)
\(L(\frac12)\) \(\approx\) \(2.61177 + 0.484438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.24 + 3.38i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.98 - 1.14i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.31 + 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.31iT - 5T^{2} \)
7 \( 1 + (3.37 - 1.94i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.635 + 0.366i)T + (5.5 + 9.52i)T^{2} \)
19 \( 1 + (2.34 - 1.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.67 + 2.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.17 - 3.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.46iT - 31T^{2} \)
37 \( 1 + (4.29 + 2.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.15 + 2.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.07 - 7.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.05iT - 47T^{2} \)
53 \( 1 - 5.23T + 53T^{2} \)
59 \( 1 + (-9.40 + 5.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.39 + 9.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.40 - 3.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.13 - 2.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 - 8.81T + 79T^{2} \)
83 \( 1 + 4.05iT - 83T^{2} \)
89 \( 1 + (11.0 + 6.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.831 + 0.480i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.67875295009812788223806371687, −12.25540465551958554769525876384, −10.53773293427570735743341912351, −9.016996288384257758332432681015, −7.897281045836579629295361764945, −6.86491857486186403716774702527, −6.45507864609765612344333561096, −5.27079513137578832937321512485, −3.29143317392540206512563403337, −2.71029763940332157946766095877, 2.53989731424406611840691262310, 3.80164139146361371616322878595, 4.24147621072340207151984146266, 5.39940629131762269518288061872, 6.85241697405602191721795988725, 8.677847492920502827270018248416, 9.668816479819998524954049836876, 10.27916750963023639477297436971, 11.31261649095583972189622995720, 12.41634615468362662744811156285

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