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Properties

Label 1-2205-2205.1609-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.799 + 0.600i$
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  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.623 + 0.781i)8-s + (0.955 − 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.365 − 0.930i)16-s + (−0.900 + 0.433i)17-s − 19-s + (−0.826 + 0.563i)22-s + (−0.826 + 0.563i)23-s + (0.900 + 0.433i)26-s + (0.826 + 0.563i)29-s + (0.5 − 0.866i)31-s + (−0.0747 + 0.997i)32-s + (0.733 − 0.680i)34-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.623 + 0.781i)8-s + (0.955 − 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.365 − 0.930i)16-s + (−0.900 + 0.433i)17-s − 19-s + (−0.826 + 0.563i)22-s + (−0.826 + 0.563i)23-s + (0.900 + 0.433i)26-s + (0.826 + 0.563i)29-s + (0.5 − 0.866i)31-s + (−0.0747 + 0.997i)32-s + (0.733 − 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.799 + 0.600i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09053282981 + 0.2710553006i\)
\(L(\frac12)\) \(\approx\) \(0.09053282981 + 0.2710553006i\)
\(L(1)\) \(\approx\) \(0.6250174882 + 0.03700316947i\)
\(L(1)\) \(\approx\) \(0.6250174882 + 0.03700316947i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.955 + 0.294i)T \)
11 \( 1 + (0.955 - 0.294i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (0.826 + 0.563i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
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

−19.30393950072114257273535409845, −18.60135511664898376809000345725, −17.759340994868931789044390248431, −17.217668763137632492828430021202, −16.594777935288017244321278339659, −15.84033333628686121561847408153, −15.025806868246842552570128000881, −14.31505412800183817467096858412, −13.35297733927051440073503069771, −12.38498685988496300175725674474, −11.83533077838583748136878514426, −11.24686571497170638480983996050, −10.17322803421739955425629274557, −9.80825421076966636011855712345, −8.74328191548613911503488728191, −8.47940285177670289610491386261, −7.22937639684067076123036860209, −6.75123917607865750306189466628, −6.04260106581825008423175277117, −4.534583550303613413689146753530, −4.06594654392363105488463126460, −2.727064654373764770913616826991, −2.14801045633424208583226604881, −1.17259339170428397846645325783, −0.08485630126939768035956606366, 0.808746988839743413231864613628, 1.88864870727430063557624982036, 2.61385553975245191937365939496, 3.79552690388211976005107758043, 4.75145637855423953989074562698, 5.88858633361792051731305195092, 6.37977115543984176440170323010, 7.23869866744263041219519673117, 8.04200644788396707371428880579, 8.72966715931755932663802798801, 9.40969088478734482396513508516, 10.23639497881907793822023726312, 10.83700832977954007842844308968, 11.7002892486153301490455959117, 12.3267009612463298794863554807, 13.344974635094312179884519560190, 14.285326899421906644939909731056, 14.97706523926462966475233204854, 15.51901242159821848982693684211, 16.40222751592430224673900382230, 17.13726609639715067759741750421, 17.538931847446498500399327209694, 18.30636165064377331249223471406, 19.19105701686240172303995537835, 19.79612829418694657205268688292

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