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Properties

Label 1-2205-2205.1352-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.957 - 0.289i$
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  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.433 − 0.900i)8-s + (−0.365 + 0.930i)11-s + (0.149 + 0.988i)13-s + (0.0747 + 0.997i)16-s + (−0.974 + 0.222i)17-s − 19-s + (0.680 − 0.733i)22-s + (0.680 − 0.733i)23-s + (0.222 − 0.974i)26-s + (−0.733 + 0.680i)29-s + (−0.5 + 0.866i)31-s + (0.294 − 0.955i)32-s + (0.988 + 0.149i)34-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.433 − 0.900i)8-s + (−0.365 + 0.930i)11-s + (0.149 + 0.988i)13-s + (0.0747 + 0.997i)16-s + (−0.974 + 0.222i)17-s − 19-s + (0.680 − 0.733i)22-s + (0.680 − 0.733i)23-s + (0.222 − 0.974i)26-s + (−0.733 + 0.680i)29-s + (−0.5 + 0.866i)31-s + (0.294 − 0.955i)32-s + (0.988 + 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002285126314 + 0.01545663002i\)
\(L(\frac12)\) \(\approx\) \(0.002285126314 + 0.01545663002i\)
\(L(1)\) \(\approx\) \(0.5587700444 + 0.01057745721i\)
\(L(1)\) \(\approx\) \(0.5587700444 + 0.01057745721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.930 - 0.365i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (0.149 + 0.988i)T \)
17 \( 1 + (-0.974 + 0.222i)T \)
19 \( 1 - T \)
23 \( 1 + (0.680 - 0.733i)T \)
29 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.974 - 0.222i)T \)
41 \( 1 + (-0.0747 + 0.997i)T \)
43 \( 1 + (0.997 - 0.0747i)T \)
47 \( 1 + (-0.930 - 0.365i)T \)
53 \( 1 + (-0.974 - 0.222i)T \)
59 \( 1 + (0.826 - 0.563i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (-0.781 - 0.623i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.149 - 0.988i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−19.07724296238933382007312605137, −18.746815433875264848009718781133, −17.72878769409217406762587474775, −17.34780612041220931880151799809, −16.502513823152668353693157756395, −15.787338302636246711820981574110, −15.20300366879151024686840314896, −14.55278132331987086139282138575, −13.4123200440865004120261514845, −12.97496290939028054631195473477, −11.720013349307437395087366154883, −10.95953431281821441307824554428, −10.662071550461534640511739000959, −9.54041269430776908144382708815, −8.990085387598636520900986867534, −8.11298524083937908445073467728, −7.641738511994192848052336418068, −6.62471560860942191471217962826, −5.89836618734963203357272838337, −5.27789412834382343560346941078, −4.07303825677617053172029117637, −2.92551169048967450878868078242, −2.21030652050613889758844058731, −1.02116176020848934725793366444, −0.007683080531474127612329461579, 1.542156183768426393194348592678, 2.1107948462907723559091937362, 3.02826123985879461718435587408, 4.15032867843760518601333851601, 4.77869549799327904532195746020, 6.22319861620876105137694969969, 6.80626493747137535105139047025, 7.509019231322816477020041944748, 8.46317222856028355481118296283, 9.07957656947490554314379167362, 9.703343423677962382473185620765, 10.76144119093361164582305634535, 11.00569626013592975983776789797, 12.030705748913400732948120177354, 12.76550846409604981751803882675, 13.25833172750084452588934070399, 14.63306525308730821477934727884, 15.058779329208288952286127953123, 16.098396171079305639598610614512, 16.56454962591797448850560640696, 17.40236751621623980924918059856, 18.01155275602777549686439794907, 18.64204462194906412590244541296, 19.413111142166734828331443253821, 19.99320682275949472568230185294

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