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Properties

Label 1-2205-2205.209-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.311 - 0.950i$
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  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.222 + 0.974i)8-s + (−0.826 − 0.563i)11-s + (0.0747 − 0.997i)13-s + (−0.733 + 0.680i)16-s + (−0.623 − 0.781i)17-s − 19-s + (−0.365 − 0.930i)22-s + (0.365 + 0.930i)23-s + (0.623 − 0.781i)26-s + (−0.365 + 0.930i)29-s + (0.5 − 0.866i)31-s + (−0.988 + 0.149i)32-s + (−0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (−0.222 + 0.974i)8-s + (−0.826 − 0.563i)11-s + (0.0747 − 0.997i)13-s + (−0.733 + 0.680i)16-s + (−0.623 − 0.781i)17-s − 19-s + (−0.365 − 0.930i)22-s + (0.365 + 0.930i)23-s + (0.623 − 0.781i)26-s + (−0.365 + 0.930i)29-s + (0.5 − 0.866i)31-s + (−0.988 + 0.149i)32-s + (−0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9228367533 - 0.6684733581i\)
\(L(\frac12)\) \(\approx\) \(0.9228367533 - 0.6684733581i\)
\(L(1)\) \(\approx\) \(1.250985674 + 0.2621898664i\)
\(L(1)\) \(\approx\) \(1.250985674 + 0.2621898664i\)

Euler product

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

−19.97727599279658260070649485584, −19.166314876629791384081896139821, −18.76901744633705142877733479880, −17.80522428658443008759413204775, −16.95950733071540768471841935823, −16.07408767483091533535386011724, −15.32671645268585065794469980781, −14.786515802725410609672080074559, −14.00488454962434575949753987106, −13.16759929642962110648609827033, −12.72533241106793967357822865205, −11.91981339651316413946730883037, −11.12023143811118261699232634259, −10.480809270325545367393448845916, −9.81910125769754785166585908174, −8.85634706663798006061621842015, −8.02089302298417819852747313944, −6.735716929786717958983645376034, −6.47405622170973756533058714023, −5.35723401621101027769118171274, −4.50309931108142728295060118473, −4.09865706103817310506155244778, −2.86678107605312371075189044957, −2.19949397942122947859858028872, −1.3703834592993622936030159914, 0.2543374040236566483762382559, 1.9716212248180838576494063333, 2.848918458425107343737800239392, 3.54201722194672989435719662325, 4.4981753271511554791222328965, 5.41561602756870186881093638448, 5.76459911924037479322984345878, 6.920686425642071519188363839882, 7.44698448971188482010040658433, 8.38623856860289819086916775340, 8.89623077183820505993768123489, 10.19173213836749304488538325433, 10.93819146480422853616241215073, 11.60981257365709637667586021356, 12.54604201525256791606375419288, 13.253356938320941269945716584621, 13.58290642869447785749405543746, 14.62198815995211391705804946768, 15.28248262083305382295938828567, 15.83038931328444403552935857109, 16.50748438287607893049074030325, 17.408815371987188421234698179787, 17.90566315662741009347115810632, 18.82310315758281310621832604140, 19.72843044656592758586901692463

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