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Properties

Label 1-261-261.241-r0-0-0
Degree $1$
Conductor $261$
Sign $0.835 + 0.549i$
Analytic cond. $1.21207$
Root an. cond. $1.21207$
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 + (0.365 + 0.930i)5-s + (0.955 − 0.294i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.826 − 0.563i)11-s + (0.0747 − 0.997i)13-s + (0.988 − 0.149i)14-s + (0.826 + 0.563i)16-s − 17-s + (0.222 + 0.974i)19-s + (0.0747 + 0.997i)20-s + (−0.733 − 0.680i)22-s + (−0.988 + 0.149i)23-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (0.955 + 0.294i)4-s + (0.365 + 0.930i)5-s + (0.955 − 0.294i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.826 − 0.563i)11-s + (0.0747 − 0.997i)13-s + (0.988 − 0.149i)14-s + (0.826 + 0.563i)16-s − 17-s + (0.222 + 0.974i)19-s + (0.0747 + 0.997i)20-s + (−0.733 − 0.680i)22-s + (−0.988 + 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(261\)    =    \(3^{2} \cdot 29\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(1.21207\)
Root analytic conductor: \(1.21207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{261} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 261,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.368411631 + 0.7089029463i\)
\(L(\frac12)\) \(\approx\) \(2.368411631 + 0.7089029463i\)
\(L(1)\) \(\approx\) \(1.970643171 + 0.3884165408i\)
\(L(1)\) \(\approx\) \(1.970643171 + 0.3884165408i\)

Euler product

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

−25.52142043585314107831735767560, −24.504443336565563846169121987286, −24.0293134124972703967544386480, −23.26331916419678163919770989759, −21.743230563293900364058754399536, −21.483673591755759226355213291380, −20.39239735037114835769870511780, −19.877166138278256407451736260087, −18.32573507153267303247690963250, −17.41963623385884925503884820683, −16.210967601174117336938945810, −15.50274603170409023053731959893, −14.3766507740697368568763828884, −13.53286870067605505231715970388, −12.67312888583416883672857822690, −11.74167096384382880090671488884, −10.87265362635513671853487035675, −9.54864238616624186126820703984, −8.390331572869251032754664458703, −7.18244523289785091402827975225, −5.90855173824866154787551004145, −4.83001733117155532632587763592, −4.37029982852046198530255253628, −2.45497728070173213334491217576, −1.62333475879290885030997681143, 1.90693828799635654193194416518, 2.95954354225862886817131199363, 4.11878292107074336904888214828, 5.42553887977618817014268620751, 6.15997903684231448270912884344, 7.50497706921280505783786956659, 8.14074106676106258707569329789, 10.16674823628253717996320506193, 10.89131043475626471495114338466, 11.680954856602841926729010337135, 13.09308591837832809733973929267, 13.76877716469908229597570844251, 14.69543590804486024267520621457, 15.37758528517562963088133035903, 16.49015095279693313173964960602, 17.72032738377084705400959125611, 18.381408088213508982525397187686, 19.83189790918339743736675232833, 20.72305517861446719800575885972, 21.50601154670567244539810886380, 22.35807645439827389537369751254, 23.13589910853466231575686923018, 24.11118396093676647657895473765, 24.81451906029985316953969991783, 25.908432463388728420425199624235

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