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Properties

Label 1-2205-2205.1663-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.150 - 0.988i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (0.781 + 0.623i)8-s + (−0.733 − 0.680i)11-s + (0.680 − 0.733i)13-s + (0.623 + 0.781i)16-s + (0.563 − 0.826i)17-s + (0.5 − 0.866i)19-s + (−0.563 − 0.826i)22-s + (−0.997 + 0.0747i)23-s + (0.826 − 0.563i)26-s + (−0.826 − 0.563i)29-s + 31-s + (0.433 + 0.900i)32-s + (0.733 − 0.680i)34-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (0.781 + 0.623i)8-s + (−0.733 − 0.680i)11-s + (0.680 − 0.733i)13-s + (0.623 + 0.781i)16-s + (0.563 − 0.826i)17-s + (0.5 − 0.866i)19-s + (−0.563 − 0.826i)22-s + (−0.997 + 0.0747i)23-s + (0.826 − 0.563i)26-s + (−0.826 − 0.563i)29-s + 31-s + (0.433 + 0.900i)32-s + (0.733 − 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.150 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.020402894 - 2.351268679i\)
\(L(\frac12)\) \(\approx\) \(2.020402894 - 2.351268679i\)
\(L(1)\) \(\approx\) \(1.815945201 - 0.1007945502i\)
\(L(1)\) \(\approx\) \(1.815945201 - 0.1007945502i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.974 + 0.222i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.680 - 0.733i)T \)
17 \( 1 + (0.563 - 0.826i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.997 + 0.0747i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.997 - 0.0747i)T \)
41 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (0.930 + 0.365i)T \)
47 \( 1 + (0.974 + 0.222i)T \)
53 \( 1 + (-0.997 + 0.0747i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.680 - 0.733i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.680 - 0.733i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
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

−19.98450892712383099036576241667, −18.97940088980603428839809623504, −18.62341557835976599120878462116, −17.52747754242602743257201820748, −16.68796085233721706131748139118, −15.86859519331518139391907198113, −15.480666051657223573660116297669, −14.43147598930909444510282377809, −14.0330941533498976868732121153, −13.21200878592910478456521682322, −12.35246302229904103240872313944, −12.03996001231183111792031244150, −10.97040770369185145334911242056, −10.36467886072016904533967993163, −9.700769175524658077169658813934, −8.53487917528188320364073070474, −7.629478619705804595495962973359, −6.974792637840273220461690913476, −5.89280463334907926341807692089, −5.55740727004742731369941287684, −4.36887379003317697008072461501, −3.88112967370688829861627192514, −2.92678098565954905073951285104, −1.9297612024076535210727021026, −1.29907344676347589392225686691, 0.32350930321305267819812781267, 1.473676790514449972940464274455, 2.77694799624974909328154825348, 3.10953445874577826163955469134, 4.17400797658778056803347820460, 5.025853141620563688904329362701, 5.76812529519787548091012713320, 6.29621282622413792189393554454, 7.53115404101581583947760979612, 7.83192524190701510157903485332, 8.84164411950002566909304326230, 9.92569204766817580351433863475, 10.7868615314280285042459130123, 11.37552110285083755943954533171, 12.15311744017405390284367352280, 12.94154802410766854038566627997, 13.705164099872228834302905689692, 13.98763499298475868755251712551, 15.08463905645695347462961852915, 15.86621391380349295619023207339, 16.01455347929843410764239837897, 17.086496425686570641825698661769, 17.84850116184354245746662151806, 18.63716871895992761405741769397, 19.49401668703513294837686257219

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