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Properties

Label 2-2200-440.107-c0-0-4
Degree $2$
Conductor $2200$
Sign $0.963 + 0.267i$
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  + (0.156 − 0.987i)2-s + (1.77 + 0.903i)3-s + (−0.951 − 0.309i)4-s + (1.16 − 1.60i)6-s + (−0.453 + 0.891i)8-s + (1.73 + 2.39i)9-s + (−0.978 − 0.207i)11-s + (−1.40 − 1.40i)12-s + (0.809 + 0.587i)16-s + (1.32 − 0.209i)17-s + (2.63 − 1.34i)18-s + (−0.128 − 0.395i)19-s + (−0.358 + 0.933i)22-s + (−1.60 + 1.16i)24-s + (0.608 + 3.84i)27-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (1.77 + 0.903i)3-s + (−0.951 − 0.309i)4-s + (1.16 − 1.60i)6-s + (−0.453 + 0.891i)8-s + (1.73 + 2.39i)9-s + (−0.978 − 0.207i)11-s + (−1.40 − 1.40i)12-s + (0.809 + 0.587i)16-s + (1.32 − 0.209i)17-s + (2.63 − 1.34i)18-s + (−0.128 − 0.395i)19-s + (−0.358 + 0.933i)22-s + (−1.60 + 1.16i)24-s + (0.608 + 3.84i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.963 + 0.267i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.963 + 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.060321205\)
\(L(\frac12)\) \(\approx\) \(2.060321205\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 \)
11 \( 1 + (0.978 + 0.207i)T \)
good3 \( 1 + (-1.77 - 0.903i)T + (0.587 + 0.809i)T^{2} \)
7 \( 1 + (0.587 - 0.809i)T^{2} \)
13 \( 1 + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (-1.32 + 0.209i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.128 + 0.395i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.773 + 0.251i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
47 \( 1 + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (1.05 + 1.05i)T + iT^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.62 - 0.829i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.285 + 1.80i)T + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + 0.209iT - T^{2} \)
97 \( 1 + (1.16 + 0.183i)T + (0.951 + 0.309i)T^{2} \)
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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

−9.287946926020203500020304045874, −8.644892724219023479545517069353, −8.007458802478457641228075620878, −7.35352183491628461074383426174, −5.59266905718974858291329735411, −4.86220714624697727623691204511, −4.07150190441229464801518195958, −3.18012249651234767587458685034, −2.74953874885331851164548922061, −1.69430837632127679128337299794, 1.36834428707780238775983956197, 2.71142275983868142373612651864, 3.41915016900164818417683331908, 4.31135705787261509833130662940, 5.48213175441471607284897629987, 6.43382648780057059519193712637, 7.18231258589135744811993274109, 7.929568012331231898076326105716, 8.100954953962051939128974540341, 8.949526059287912999275125234743

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