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Properties

Label 2-2200-440.307-c0-0-6
Degree $2$
Conductor $2200$
Sign $0.189 + 0.981i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (1.22 − 1.22i)3-s − 1.00i·4-s + 1.73i·6-s + (0.707 + 0.707i)8-s − 1.99i·9-s + (−0.5 − 0.866i)11-s + (−1.22 − 1.22i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + (1.41 + 1.41i)18-s − 1.73·19-s + (0.965 + 0.258i)22-s + 1.73·24-s + (−1.22 − 1.22i)27-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (1.22 − 1.22i)3-s − 1.00i·4-s + 1.73i·6-s + (0.707 + 0.707i)8-s − 1.99i·9-s + (−0.5 − 0.866i)11-s + (−1.22 − 1.22i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + (1.41 + 1.41i)18-s − 1.73·19-s + (0.965 + 0.258i)22-s + 1.73·24-s + (−1.22 − 1.22i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134251326\)
\(L(\frac12)\) \(\approx\) \(1.134251326\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
7 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
19 \( 1 + 1.73T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.641845540849319259587391750650, −8.408823251289446414030347827821, −7.56061041843891731181686059526, −7.06346066629369956622084103696, −6.21890209838779291024186010857, −5.49985226408241105849265077726, −4.11897644915640759523873764558, −2.84911163445146896585810625178, −2.09409395755214121906050532648, −0.857262125595525439887593538071, 1.91732929564155872406998923531, 2.61824919359236507761904989148, 3.60330510053694086268037332883, 4.23341378380421613997028132541, 4.98618499250648228058544149850, 6.46259850925084708482330337943, 7.65360514273654555880835141233, 8.130641412306271298636709367034, 8.764495307805699600400394355276, 9.543910771883311626709790141236

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