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Properties

Label 1-2205-2205.23-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.729 + 0.684i$
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  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (0.988 + 0.149i)11-s + (0.149 − 0.988i)13-s + (−0.900 + 0.433i)16-s + (0.680 − 0.733i)17-s + (0.5 + 0.866i)19-s + (0.680 + 0.733i)22-s + (0.294 − 0.955i)23-s + (0.733 − 0.680i)26-s + (−0.733 − 0.680i)29-s + 31-s + (−0.974 − 0.222i)32-s + (0.988 − 0.149i)34-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.433 + 0.900i)8-s + (0.988 + 0.149i)11-s + (0.149 − 0.988i)13-s + (−0.900 + 0.433i)16-s + (0.680 − 0.733i)17-s + (0.5 + 0.866i)19-s + (0.680 + 0.733i)22-s + (0.294 − 0.955i)23-s + (0.733 − 0.680i)26-s + (−0.733 − 0.680i)29-s + 31-s + (−0.974 − 0.222i)32-s + (0.988 − 0.149i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.729 + 0.684i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.716243977 + 1.074902008i\)
\(L(\frac12)\) \(\approx\) \(2.716243977 + 1.074902008i\)
\(L(1)\) \(\approx\) \(1.674375481 + 0.5954105912i\)
\(L(1)\) \(\approx\) \(1.674375481 + 0.5954105912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.781 + 0.623i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (0.149 - 0.988i)T \)
17 \( 1 + (0.680 - 0.733i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.294 - 0.955i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.294 - 0.955i)T \)
41 \( 1 + (-0.0747 - 0.997i)T \)
43 \( 1 + (0.997 + 0.0747i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.294 - 0.955i)T \)
59 \( 1 + (-0.900 + 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.149 - 0.988i)T \)
79 \( 1 - T \)
83 \( 1 + (0.149 + 0.988i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
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

−19.760818344245692549340184853148, −19.0354239358494491252383457479, −18.57737346597157190585145426122, −17.39694512261103914197417915974, −16.79627855792686372104703131124, −15.835315507564024627984565011, −15.171209399898148829393441558815, −14.376628055661068487348145879641, −13.839206798663229658373681921533, −13.15465147747573662295233872317, −12.23971743339725251707272196377, −11.646632278539601340205996193628, −11.11696024173795051224036904416, −10.17400051284947071650087890696, −9.3723389972091019690034811987, −8.84851921355960040983273903002, −7.531635412582987770771302062770, −6.69021829136665982302003518713, −6.05388872286169276491381453229, −5.15402369062307638384913280957, −4.32632329334777664784653921213, −3.610181504250027390710542825237, −2.85202590470899117901630173445, −1.674726454952095997720420815206, −1.1095483531953579888733177102, 0.878410806623264942310766929864, 2.22612082047491038069923537746, 3.14978704345194869535559945644, 3.873109949238137078879373609465, 4.6782544332830163320703194189, 5.65235474235732782390197319141, 6.08817147267786961355968844974, 7.16021180950381968678001243571, 7.66897667547775437035570157202, 8.55079331710009009244713340373, 9.34376374253133265408476689910, 10.3051367733294613547289629970, 11.22194688600455162225819809463, 12.135661948283748123881811976423, 12.42895013133582550453476232488, 13.43219595462995285998802985635, 14.15277339548449435791493944034, 14.63279170686301560764581533900, 15.46429918866327624939068660667, 16.115249553740641957511511561707, 16.86508969717562924074984438371, 17.43091351296180581622138842459, 18.23662283723065908891154835542, 19.070725617724236632932861592670, 20.04627226480989244880549131453

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