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Properties

Label 2-220-220.219-c1-0-14
Degree $2$
Conductor $220$
Sign $0.436 - 0.899i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
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  + (0.419 + 1.35i)2-s + 2.05·3-s + (−1.64 + 1.13i)4-s + (1.54 − 1.61i)5-s + (0.860 + 2.77i)6-s + 1.06i·7-s + (−2.22 − 1.75i)8-s + 1.21·9-s + (2.83 + 1.40i)10-s + (3.17 + 0.957i)11-s + (−3.38 + 2.32i)12-s − 4.77·13-s + (−1.43 + 0.445i)14-s + (3.16 − 3.32i)15-s + (1.43 − 3.73i)16-s + 0.329·17-s + ⋯
L(s)  = 1  + (0.296 + 0.955i)2-s + 1.18·3-s + (−0.824 + 0.566i)4-s + (0.689 − 0.724i)5-s + (0.351 + 1.13i)6-s + 0.401i·7-s + (−0.785 − 0.619i)8-s + 0.404·9-s + (0.896 + 0.443i)10-s + (0.957 + 0.288i)11-s + (−0.976 + 0.670i)12-s − 1.32·13-s + (−0.383 + 0.119i)14-s + (0.817 − 0.858i)15-s + (0.358 − 0.933i)16-s + 0.0799·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61083 + 1.00882i\)
\(L(\frac12)\) \(\approx\) \(1.61083 + 1.00882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.419 - 1.35i)T \)
5 \( 1 + (-1.54 + 1.61i)T \)
11 \( 1 + (-3.17 - 0.957i)T \)
good3 \( 1 - 2.05T + 3T^{2} \)
7 \( 1 - 1.06iT - 7T^{2} \)
13 \( 1 + 4.77T + 13T^{2} \)
17 \( 1 - 0.329T + 17T^{2} \)
19 \( 1 + 0.677T + 19T^{2} \)
23 \( 1 + 4.71T + 23T^{2} \)
29 \( 1 + 7.19iT - 29T^{2} \)
31 \( 1 - 9.04iT - 31T^{2} \)
37 \( 1 + 5.33iT - 37T^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 + 5.08iT - 43T^{2} \)
47 \( 1 - 1.61T + 47T^{2} \)
53 \( 1 - 8.57iT - 53T^{2} \)
59 \( 1 + 0.336iT - 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 - 4.71T + 67T^{2} \)
71 \( 1 + 3.97iT - 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 6.35T + 79T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 + 7.95T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−12.60556186694575614606363142233, −12.07144947415463049357999350254, −9.857126535748668803930809362278, −9.273019478327502925838701349160, −8.512653955595243262341014981165, −7.58380625328105966610486994457, −6.32721262493830750041292175418, −5.15526062085783509312008189400, −3.97476532787784431356844495100, −2.35474635857878699814471023314, 2.00256857398720644735119045098, 3.03814282720150971055099651533, 4.13643087633955919614240619822, 5.74370037527311643300209060761, 7.12750384736812723936130074949, 8.447982809422279294836955647977, 9.511416048406765607561663978870, 9.979365810440346592719634932265, 11.12892343233075515735327989610, 12.12318173816898183332504722816

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