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Properties

Label 1-275-275.169-r0-0-0
Degree $1$
Conductor $275$
Sign $0.920 + 0.390i$
Analytic cond. $1.27709$
Root an. cond. $1.27709$
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  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 6-s + (0.809 − 0.587i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)12-s + (−0.309 + 0.951i)13-s + 14-s + (−0.809 + 0.587i)16-s − 17-s + (0.809 − 0.587i)18-s + (0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 6-s + (0.809 − 0.587i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)12-s + (−0.309 + 0.951i)13-s + 14-s + (−0.809 + 0.587i)16-s − 17-s + (0.809 − 0.587i)18-s + (0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(1.27709\)
Root analytic conductor: \(1.27709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 275,\ (0:\ ),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.521967175 + 0.5122324772i\)
\(L(\frac12)\) \(\approx\) \(2.521967175 + 0.5122324772i\)
\(L(1)\) \(\approx\) \(2.036339299 + 0.3401363140i\)
\(L(1)\) \(\approx\) \(2.036339299 + 0.3401363140i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 - T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 - T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 - T \)
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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

−25.21786044980413847018039791903, −24.83653123859577705174868570727, −23.84026303187058246519866213953, −22.51536889498533108817259125279, −21.98654699088155275805520215160, −20.89730922287781912793429459736, −20.53942971502934735295041015894, −19.48186240391218926835003296595, −18.65572299541812722429240302760, −17.42904721027104743021563893582, −15.83346151401885535922831729307, −15.2175531987638657387622021223, −14.48964420923863127814600787767, −13.57099368372419203841566148838, −12.619468993248313500850192498123, −11.461861287535730061508580697077, −10.59218884758323817022447215602, −9.60695143655020744736356060922, −8.561583226066083492617441099, −7.404562282130138461818364362011, −5.70723048189045211136482870811, −4.897297663211957962732040022399, −3.82016966248543524710320973539, −2.72075336174460711870809619229, −1.75521631551684786204775237241, 1.721752331029730347278695736194, 2.902511003608236813116616299735, 4.16044569183064386731951186605, 5.03735319090151925985498131441, 6.81988338979459681231537117623, 7.07845681122535211271161653362, 8.36904409494478871758104021275, 9.08884859655609670063570567149, 10.919753593490390643524818939836, 11.889821589084940189949494856242, 12.980248465416368189336147114686, 13.75383209935334329145899292955, 14.46734834488366463708024956819, 15.21707630590958220546543160864, 16.39819264145642951990216725920, 17.46730563852642760238759453619, 18.21358023396145740306962399095, 19.617215272652360472500752920038, 20.3615115002909237305932910160, 21.235659864877915005100767241260, 22.09095678822386076603818694084, 23.44088592832045596722083395121, 23.993131746941620634733286556002, 24.62914297088344096728160550336, 25.55235010210241999217118683135

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