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Properties

Label 2-2205-105.44-c0-0-2
Degree $2$
Conductor $2205$
Sign $0.994 - 0.107i$
Analytic cond. $1.10043$
Root an. cond. $1.04901$
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  + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.22 − 0.707i)11-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)19-s − 0.999i·20-s + (0.499 + 0.866i)25-s − 1.41i·29-s + (0.707 + 1.22i)31-s + 2i·41-s + (1.22 + 0.707i)44-s − 1.41·55-s + (1.73 − i)59-s + (0.707 − 1.22i)61-s − 0.999·64-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (1.22 − 0.707i)11-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)19-s − 0.999i·20-s + (0.499 + 0.866i)25-s − 1.41i·29-s + (0.707 + 1.22i)31-s + 2i·41-s + (1.22 + 0.707i)44-s − 1.41·55-s + (1.73 − i)59-s + (0.707 − 1.22i)61-s − 0.999·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.994 - 0.107i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ 0.994 - 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237838520\)
\(L(\frac12)\) \(\approx\) \(1.237838520\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.039995301034037183360064824458, −8.378900757773221130452558395499, −7.86112380009057524558983776407, −6.88200939091123417031489998891, −6.44335385562245058811059321740, −5.13514757555995929290094790510, −4.22979933922522501601450036040, −3.53088605904593266922060193664, −2.69006814362129204180578256354, −1.11643982858206693492446354024, 1.20650900898250811954904043506, 2.33050738565316552121705498527, 3.55583600713291679649288099025, 4.26780171235210779505480773120, 5.36306280199290156064901929432, 6.17421902590118357204182220189, 7.02987572488431192644121155326, 7.37456527855466212701119512040, 8.464937169346138123058183859717, 9.328873975338688321235849017808

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