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Properties

Label 1-2205-2205.1192-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: $\chi_{2205} (1192, \cdot )$
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

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

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