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Properties

Label 1-2205-2205.1388-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.966 + 0.255i$
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  + (−0.294 + 0.955i)2-s + (−0.826 − 0.563i)4-s + (0.781 − 0.623i)8-s + (0.222 + 0.974i)11-s + (−0.294 + 0.955i)13-s + (0.365 + 0.930i)16-s + (0.563 + 0.826i)17-s + (0.5 + 0.866i)19-s + (−0.997 − 0.0747i)22-s + (0.433 + 0.900i)23-s + (−0.826 − 0.563i)26-s + (0.0747 + 0.997i)29-s + (−0.5 − 0.866i)31-s + (−0.997 + 0.0747i)32-s + (−0.955 + 0.294i)34-s + ⋯
L(s)  = 1  + (−0.294 + 0.955i)2-s + (−0.826 − 0.563i)4-s + (0.781 − 0.623i)8-s + (0.222 + 0.974i)11-s + (−0.294 + 0.955i)13-s + (0.365 + 0.930i)16-s + (0.563 + 0.826i)17-s + (0.5 + 0.866i)19-s + (−0.997 − 0.0747i)22-s + (0.433 + 0.900i)23-s + (−0.826 − 0.563i)26-s + (0.0747 + 0.997i)29-s + (−0.5 − 0.866i)31-s + (−0.997 + 0.0747i)32-s + (−0.955 + 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.966 + 0.255i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.966 + 0.255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1470140046 + 1.133594922i\)
\(L(\frac12)\) \(\approx\) \(0.1470140046 + 1.133594922i\)
\(L(1)\) \(\approx\) \(0.6824115826 + 0.5507974455i\)
\(L(1)\) \(\approx\) \(0.6824115826 + 0.5507974455i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.294 + 0.955i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.294 + 0.955i)T \)
17 \( 1 + (0.563 + 0.826i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 + (0.0747 + 0.997i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.997 - 0.0747i)T \)
41 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (0.149 - 0.988i)T \)
47 \( 1 + (-0.294 + 0.955i)T \)
53 \( 1 + (-0.997 - 0.0747i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.680 - 0.733i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.294 - 0.955i)T \)
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

−19.576682330564650552987720814263, −18.673194359006865955441025235763, −18.14461966369420367462993279315, −17.43134843997666739160456195525, −16.61457298858530593614049563109, −16.02446130598337102787398607677, −14.91407852199313984514489369303, −14.123923348375221583160976750093, −13.46356907384544691949640384667, −12.74482330553079608083917917865, −12.03380060019682134765187293543, −11.18995464461959545260847349184, −10.785376135674188866329717855381, −9.74128962275203914657496025458, −9.286130456498106063891878590149, −8.302883603339241344252728504667, −7.77932257211751113369524935353, −6.75256394110159917858505743502, −5.59644534956187556553849268256, −4.92124157071200839579906872835, −3.97890159803580093153248908269, −2.91365585110183432938207975962, −2.694208369464504663145391738760, −1.18757411182073335547090302053, −0.50533228696153516797984325921, 1.247188591731081124818318848986, 1.964500169357054605599515437446, 3.50245577614113007395681393160, 4.26024755352035768515208568176, 5.06740904611821263375649871527, 5.90336294901044620757170863234, 6.62978186985107680379504789876, 7.53990279655800523379966541936, 7.864271042779794509803382998751, 9.15908570236590565841038246412, 9.43825091897314156312460870697, 10.2818833906131161359156190438, 11.168252400101904476805310149189, 12.232424368262975320286798327, 12.81614964938334047734691047435, 13.77378749727718515310646875028, 14.56762554068465085927968184616, 14.85474875277517743528324577178, 15.84277932776298132138810332711, 16.48967911697960256417400394391, 17.149590066534303618947904927008, 17.722325158259298385106647974806, 18.58990602252239130881857576179, 19.130017963602458008315908304541, 19.879388599965648386575392290487

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