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Properties

Label 1-1305-1305.562-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.839 - 0.543i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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  + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.294 + 0.955i)7-s + (−0.900 + 0.433i)8-s + (−0.563 − 0.826i)11-s + (−0.997 + 0.0747i)13-s + (0.149 − 0.988i)14-s + (0.826 − 0.563i)16-s + 17-s + (−0.974 − 0.222i)19-s + (0.680 + 0.733i)22-s + (−0.149 + 0.988i)23-s + (0.974 − 0.222i)26-s + i·28-s + (−0.930 − 0.365i)31-s + (−0.733 + 0.680i)32-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.294 + 0.955i)7-s + (−0.900 + 0.433i)8-s + (−0.563 − 0.826i)11-s + (−0.997 + 0.0747i)13-s + (0.149 − 0.988i)14-s + (0.826 − 0.563i)16-s + 17-s + (−0.974 − 0.222i)19-s + (0.680 + 0.733i)22-s + (−0.149 + 0.988i)23-s + (0.974 − 0.222i)26-s + i·28-s + (−0.930 − 0.365i)31-s + (−0.733 + 0.680i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (562, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6127037866 - 0.1811783636i\)
\(L(\frac12)\) \(\approx\) \(0.6127037866 - 0.1811783636i\)
\(L(1)\) \(\approx\) \(0.6035914908 + 0.02726418453i\)
\(L(1)\) \(\approx\) \(0.6035914908 + 0.02726418453i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (-0.294 + 0.955i)T \)
11 \( 1 + (-0.563 - 0.826i)T \)
13 \( 1 + (-0.997 + 0.0747i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.974 - 0.222i)T \)
23 \( 1 + (-0.149 + 0.988i)T \)
31 \( 1 + (-0.930 - 0.365i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.365 - 0.930i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.294 - 0.955i)T \)
67 \( 1 + (0.563 - 0.826i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (0.997 + 0.0747i)T \)
83 \( 1 + (-0.680 + 0.733i)T \)
89 \( 1 + (-0.781 + 0.623i)T \)
97 \( 1 + (0.733 + 0.680i)T \)
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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

−20.898368519275907654869010471479, −19.981979985583163528899163822892, −19.705636313678265620860217410782, −18.67899234667763817062848684075, −18.06914497699186490623549628907, −17.13636680605394812525397351404, −16.7055715000758675817611097169, −16.002697806458318548832585949580, −14.84909278457400486120447731134, −14.47883446651964361183412707431, −12.91777307692075230917876305548, −12.634202930107409085046236806486, −11.5956512731163180449424552455, −10.61993387116627935137439349812, −10.070990857825808786269275825747, −9.563255220862118866396074759033, −8.34958681701745921534551291977, −7.65259132784144612982260657273, −7.03260899511746143454245729091, −6.18737296804640569201454247294, −4.93777112410765394682512196727, −3.93963539243671440727659240118, −2.84862377913057243198891690037, −2.01446401573544175973752658245, −0.81740501649238519010764409627, 0.4546938789088175232574579513, 1.946842728007986601282963926051, 2.63259886362132385892838173051, 3.60250858372461212759174895332, 5.26965724160963654725348929051, 5.73719762188859813729075648611, 6.70067116460193266863353170403, 7.67942297317460080223303747262, 8.28295299194687617157394186829, 9.233383100002072530739988593958, 9.73685464523725046210910693361, 10.705657768275201626017964336950, 11.45560134624579684470970414924, 12.26675823359939353921290657741, 12.97867754203831393571045552928, 14.27035620377563329641246295374, 14.99084653627367606410869919004, 15.66148808095752705215748730692, 16.47162153757010155349223517562, 17.024938076253464744921843067517, 18.01764266416581507808880318179, 18.5998499552140771579822475527, 19.35171771705752409881318413365, 19.72138353639695048312077288338, 21.04077646102736870929540523086

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