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Properties

Label 2-220-44.39-c1-0-10
Degree $2$
Conductor $220$
Sign $0.732 - 0.680i$
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  + (1.41 − 0.0434i)2-s + (−0.722 + 0.994i)3-s + (1.99 − 0.122i)4-s + (−0.309 + 0.951i)5-s + (−0.978 + 1.43i)6-s + (−0.948 + 0.689i)7-s + (2.81 − 0.260i)8-s + (0.460 + 1.41i)9-s + (−0.395 + 1.35i)10-s + (3.11 − 1.14i)11-s + (−1.32 + 2.07i)12-s + (1.03 − 0.336i)13-s + (−1.31 + 1.01i)14-s + (−0.722 − 0.994i)15-s + (3.96 − 0.490i)16-s + (−4.70 − 1.52i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0307i)2-s + (−0.417 + 0.574i)3-s + (0.998 − 0.0614i)4-s + (−0.138 + 0.425i)5-s + (−0.399 + 0.586i)6-s + (−0.358 + 0.260i)7-s + (0.995 − 0.0920i)8-s + (0.153 + 0.471i)9-s + (−0.125 + 0.429i)10-s + (0.938 − 0.343i)11-s + (−0.381 + 0.598i)12-s + (0.287 − 0.0933i)13-s + (−0.350 + 0.271i)14-s + (−0.186 − 0.256i)15-s + (0.992 − 0.122i)16-s + (−1.14 − 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(220\)    =    \(2^{2} \cdot 5 \cdot 11\)
Sign: $0.732 - 0.680i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{220} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 220,\ (\ :1/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76658 + 0.694105i\)
\(L(\frac12)\) \(\approx\) \(1.76658 + 0.694105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0434i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-3.11 + 1.14i)T \)
good3 \( 1 + (0.722 - 0.994i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (0.948 - 0.689i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.03 + 0.336i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.70 + 1.52i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.36 + 3.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.28iT - 23T^{2} \)
29 \( 1 + (-0.622 - 0.856i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.07 + 1.97i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.19 + 0.869i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.01 + 4.14i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 + (-0.973 + 1.34i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.39 + 4.28i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.36 + 3.25i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (13.2 + 4.30i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + (-14.9 - 4.84i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.53 - 11.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.99 - 9.20i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.989 - 3.04i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.254T + 89T^{2} \)
97 \( 1 + (0.00979 + 0.0301i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.41737736958392022232482625131, −11.26354080612769302048620573382, −10.93246929831450853452559609766, −9.767411428787498489591089508173, −8.399264050470926861237721418231, −6.85462864846835005930132113276, −6.20525778227900236882924032661, −4.84581392550468243132209204006, −3.96100825597302374629343378617, −2.47123309731214470634783270092, 1.62907234149835594352884690199, 3.65812977026697535832247971015, 4.61954457434242279951035309727, 6.28563305547982758569203007900, 6.55298536768121243794799542749, 7.88614952488553819250585682926, 9.252310460164135714029905462690, 10.54802691688676347463141852270, 11.65427253768267768946714262538, 12.24321327998089756289830703752

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