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Properties

Label 2-220-20.3-c1-0-24
Degree $2$
Conductor $220$
Sign $0.932 + 0.361i$
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  + (1.36 + 0.354i)2-s + (1.16 − 1.16i)3-s + (1.74 + 0.971i)4-s + (−1.42 − 1.72i)5-s + (2.00 − 1.18i)6-s + (−1.60 − 1.60i)7-s + (2.04 + 1.94i)8-s + 0.282i·9-s + (−1.33 − 2.86i)10-s − i·11-s + (3.17 − 0.906i)12-s + (3.94 + 3.94i)13-s + (−1.62 − 2.76i)14-s + (−3.66 − 0.350i)15-s + (2.11 + 3.39i)16-s + (−4.86 + 4.86i)17-s + ⋯
L(s)  = 1  + (0.968 + 0.250i)2-s + (0.673 − 0.673i)3-s + (0.874 + 0.485i)4-s + (−0.636 − 0.771i)5-s + (0.820 − 0.482i)6-s + (−0.605 − 0.605i)7-s + (0.724 + 0.689i)8-s + 0.0940i·9-s + (−0.423 − 0.906i)10-s − 0.301i·11-s + (0.915 − 0.261i)12-s + (1.09 + 1.09i)13-s + (−0.434 − 0.737i)14-s + (−0.947 − 0.0903i)15-s + (0.528 + 0.848i)16-s + (−1.17 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19550 - 0.411202i\)
\(L(\frac12)\) \(\approx\) \(2.19550 - 0.411202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.354i)T \)
5 \( 1 + (1.42 + 1.72i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-1.16 + 1.16i)T - 3iT^{2} \)
7 \( 1 + (1.60 + 1.60i)T + 7iT^{2} \)
13 \( 1 + (-3.94 - 3.94i)T + 13iT^{2} \)
17 \( 1 + (4.86 - 4.86i)T - 17iT^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 + (-3.01 + 3.01i)T - 23iT^{2} \)
29 \( 1 + 6.65iT - 29T^{2} \)
31 \( 1 + 0.619iT - 31T^{2} \)
37 \( 1 + (-0.294 + 0.294i)T - 37iT^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 + (-2.52 + 2.52i)T - 43iT^{2} \)
47 \( 1 + (8.42 + 8.42i)T + 47iT^{2} \)
53 \( 1 + (-2.22 - 2.22i)T + 53iT^{2} \)
59 \( 1 - 1.02T + 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 + (-3.45 - 3.45i)T + 67iT^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (-3.57 - 3.57i)T + 73iT^{2} \)
79 \( 1 + 2.91T + 79T^{2} \)
83 \( 1 + (9.90 - 9.90i)T - 83iT^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + (0.328 - 0.328i)T - 97iT^{2} \)
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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

−12.71069010546198496567667214495, −11.47690992955292426396403291367, −10.66227509835394355814480031973, −8.712103796793425688302497084998, −8.277935171543900787343441655395, −6.98984544180097527125053584934, −6.25487380818414382055181071190, −4.48494222434293962873903504024, −3.72191709984007703215726337286, −1.97311584535653205051091252422, 2.77313039237660028503755075501, 3.44169427168156585695751272516, 4.61417905784308438211320886950, 6.13359429410784112734441486489, 7.02131983374445992534478412972, 8.481022276581061582180184028137, 9.556758191823884819699795100642, 10.66731757326797496645782873636, 11.30154555376606884240928756559, 12.49444040677178681716059941197

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