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Properties

Label 1-1305-1305.727-r0-0-0
Degree $1$
Conductor $1305$
Sign $0.0144 - 0.999i$
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  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.563 + 0.826i)7-s + (0.623 − 0.781i)8-s + (−0.930 − 0.365i)11-s + (−0.149 − 0.988i)13-s + (−0.294 + 0.955i)14-s + (0.365 − 0.930i)16-s + 17-s + (0.433 − 0.900i)19-s + (−0.997 − 0.0747i)22-s + (0.294 − 0.955i)23-s + (−0.433 − 0.900i)26-s + i·28-s + (−0.680 + 0.733i)31-s + (0.0747 − 0.997i)32-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (−0.563 + 0.826i)7-s + (0.623 − 0.781i)8-s + (−0.930 − 0.365i)11-s + (−0.149 − 0.988i)13-s + (−0.294 + 0.955i)14-s + (0.365 − 0.930i)16-s + 17-s + (0.433 − 0.900i)19-s + (−0.997 − 0.0747i)22-s + (0.294 − 0.955i)23-s + (−0.433 − 0.900i)26-s + i·28-s + (−0.680 + 0.733i)31-s + (0.0747 − 0.997i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0144 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0144 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $0.0144 - 0.999i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ 0.0144 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.684729703 - 1.709278276i\)
\(L(\frac12)\) \(\approx\) \(1.684729703 - 1.709278276i\)
\(L(1)\) \(\approx\) \(1.595160131 - 0.5771438008i\)
\(L(1)\) \(\approx\) \(1.595160131 - 0.5771438008i\)

Euler product

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

−21.10989831914312222596062347607, −20.69975664373053451165515238800, −19.785027335903085867719622048280, −19.05095227691561409543652926488, −18.06120362136231875455819149434, −16.99128596730466508986622864635, −16.47161241323082909295403370946, −15.878697664732228859196896414759, −14.89346983557746185942509551554, −14.2152684349140899620813078643, −13.472300239953259742982100952456, −12.86184577735402542984052367095, −12.02736550174537469196206499909, −11.265682003993190293743168630361, −10.26853228866795622459744933224, −9.65538872007480405384904579580, −8.287883143619638496649464756748, −7.35921489568861030482407638256, −7.03395192446531820384156461290, −5.81565840147943057710609559751, −5.23137453186964713235297184243, −4.087301064066800659248916839618, −3.55331367244341443630926215870, −2.50384445780515140102406477505, −1.40924073103155577679297727008, 0.65907447006668571710706445659, 2.123506684482410055472180419567, 3.00102644848795122886364395136, 3.42150516779018395148769441623, 4.98031809967993219020566934493, 5.33040275691857759701812761778, 6.20186254302615065865192161349, 7.127377297432594789409681251668, 8.06494389034481814005792374587, 9.094499824129360900735853362642, 10.163509419016455898607062348998, 10.64566504425790032481890898733, 11.67603003865298614293972713069, 12.47621195445708625922111029036, 12.92115341554118618703645802219, 13.75098935640127828921731175495, 14.6394091945465117482386612927, 15.436559542060271020521448072667, 15.880422624366761481502339084077, 16.70484828218926171840309873038, 17.90543380638140836367356850072, 18.754876100740674533876210890987, 19.25807261750413782568678134390, 20.27669004569326994204082683155, 20.80556747598305461502449476318

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