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Properties

Label 1-2205-2205.2092-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.972 + 0.234i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (0.563 + 0.826i)2-s + (−0.365 + 0.930i)4-s + (−0.974 + 0.222i)8-s + (0.826 − 0.563i)11-s + (−0.997 + 0.0747i)13-s + (−0.733 − 0.680i)16-s + (−0.781 − 0.623i)17-s + 19-s + (0.930 + 0.365i)22-s + (−0.930 − 0.365i)23-s + (−0.623 − 0.781i)26-s + (−0.365 − 0.930i)29-s + (0.5 + 0.866i)31-s + (0.149 − 0.988i)32-s + (0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (0.563 + 0.826i)2-s + (−0.365 + 0.930i)4-s + (−0.974 + 0.222i)8-s + (0.826 − 0.563i)11-s + (−0.997 + 0.0747i)13-s + (−0.733 − 0.680i)16-s + (−0.781 − 0.623i)17-s + 19-s + (0.930 + 0.365i)22-s + (−0.930 − 0.365i)23-s + (−0.623 − 0.781i)26-s + (−0.365 − 0.930i)29-s + (0.5 + 0.866i)31-s + (0.149 − 0.988i)32-s + (0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2092, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.972 + 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688428953 + 0.2006487047i\)
\(L(\frac12)\) \(\approx\) \(1.688428953 + 0.2006487047i\)
\(L(1)\) \(\approx\) \(1.173490562 + 0.4412900591i\)
\(L(1)\) \(\approx\) \(1.173490562 + 0.4412900591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.563 + 0.826i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (-0.997 + 0.0747i)T \)
17 \( 1 + (-0.781 - 0.623i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.930 - 0.365i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.781 + 0.623i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.680 + 0.733i)T \)
47 \( 1 + (-0.563 - 0.826i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + (0.955 - 0.294i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.997 + 0.0747i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.91283088946185402979335219367, −19.30623311510934808092301108932, −18.262652865501562781871595132587, −17.76419129600744448287502063629, −16.93608879922479280368360647160, −15.95422128413449739988506019543, −15.05958163759514181064366170543, −14.601498534507161939116634922169, −13.85088446427619267605642869058, −13.05055920497875040007472891554, −12.3589386746190241758810763003, −11.72931785281063441790542332538, −11.10153356292284455716578446361, −10.0832979327058014508525627890, −9.61059565447453228738777932804, −8.89079529641109536525087153219, −7.74526362220734644482108628657, −6.86355426599742806212295642738, −6.01052933329600353554373396112, −5.20404011556571246966753652299, −4.32825552993648560938358052691, −3.77797833854701234028356898280, −2.67106694080228589528865124987, −1.96701903257719840638721850140, −1.01104556959067311091917563879, 0.52544100686411002265594856342, 2.11507763497171802247512674759, 3.028452040416764353342167179918, 3.896746101960507317520922814235, 4.69556260252268790634709314515, 5.3990025779678929785227223244, 6.35654003429258941933254253431, 6.88033694466503323421638619221, 7.76483501011309947107619871411, 8.452072189028953309482772254993, 9.35769365260111923633970541716, 9.915509398223782032054378398188, 11.33571780478629174306555918900, 11.80063362939634164834128739103, 12.51749511134032919950945886591, 13.52023802053146488751518153969, 13.96003199169974677570905872343, 14.67626804451749445168794310514, 15.379033398972708945996560191972, 16.24475245027693235241634898502, 16.627028246692195927351788380396, 17.57932633015685365680550762983, 18.00634025817631946637731013234, 18.991100439342555383795647219824, 19.85784616240521429614839003055

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