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Properties

Label 1-131-131.90-r1-0-0
Degree $1$
Conductor $131$
Sign $0.402 + 0.915i$
Analytic cond. $14.0779$
Root an. cond. $14.0779$
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  + (−0.981 + 0.192i)2-s + (0.215 − 0.976i)3-s + (0.926 − 0.377i)4-s + (−0.861 − 0.506i)5-s + (−0.0241 + 0.999i)6-s + (0.958 + 0.285i)7-s + (−0.836 + 0.548i)8-s + (−0.906 − 0.421i)9-s + (0.943 + 0.331i)10-s + (−0.168 + 0.985i)11-s + (−0.168 − 0.985i)12-s + (−0.943 + 0.331i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.715 − 0.698i)16-s + (0.443 + 0.896i)17-s + ⋯
L(s)  = 1  + (−0.981 + 0.192i)2-s + (0.215 − 0.976i)3-s + (0.926 − 0.377i)4-s + (−0.861 − 0.506i)5-s + (−0.0241 + 0.999i)6-s + (0.958 + 0.285i)7-s + (−0.836 + 0.548i)8-s + (−0.906 − 0.421i)9-s + (0.943 + 0.331i)10-s + (−0.168 + 0.985i)11-s + (−0.168 − 0.985i)12-s + (−0.943 + 0.331i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.715 − 0.698i)16-s + (0.443 + 0.896i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(14.0779\)
Root analytic conductor: \(14.0779\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (1:\ ),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5086304329 + 0.3319367179i\)
\(L(\frac12)\) \(\approx\) \(0.5086304329 + 0.3319367179i\)
\(L(1)\) \(\approx\) \(0.6100560337 - 0.04821946316i\)
\(L(1)\) \(\approx\) \(0.6100560337 - 0.04821946316i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (-0.981 + 0.192i)T \)
3 \( 1 + (0.215 - 0.976i)T \)
5 \( 1 + (-0.861 - 0.506i)T \)
7 \( 1 + (0.958 + 0.285i)T \)
11 \( 1 + (-0.168 + 0.985i)T \)
13 \( 1 + (-0.943 + 0.331i)T \)
17 \( 1 + (0.443 + 0.896i)T \)
19 \( 1 + (-0.885 + 0.464i)T \)
23 \( 1 + (0.262 - 0.964i)T \)
29 \( 1 + (0.906 - 0.421i)T \)
31 \( 1 + (-0.779 - 0.626i)T \)
37 \( 1 + (0.0724 + 0.997i)T \)
41 \( 1 + (0.715 + 0.698i)T \)
43 \( 1 + (0.399 + 0.916i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.989 + 0.144i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.399 + 0.916i)T \)
71 \( 1 + (0.354 - 0.935i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.748 - 0.663i)T \)
83 \( 1 + (0.527 + 0.849i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.607 + 0.794i)T \)
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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

−27.58695779636993108885455115519, −27.366523983202836576428549823175, −26.6870364486480485243337375385, −25.6501348646244802079056960412, −24.44028654192887828563305064128, −23.29325004699474604000746763504, −21.811911157703456339150027419564, −21.14403761126672055750648980273, −19.95487424048582010444416343052, −19.34759598340548907071143950279, −18.05663372404716901722870959715, −16.976831428960467575934197178655, −16.00243704382054884359612804314, −15.1154158331487195554713770121, −14.158497771934731032021318518916, −12.03481029111686743692066239037, −11.04588269547313560490637819946, −10.52178827292618957901512093700, −9.09559758349785818691164069577, −8.11567949260153832840185692214, −7.200825957251055240740730235846, −5.24560837394722122213059160567, −3.71647313519912295075240588137, −2.5958028490031556863390972121, −0.34288797822435661404172794260, 1.29934669230507123419216044927, 2.43298956232549898387788713581, 4.66204794116466345721718999171, 6.30804308464600437571931371860, 7.66857604851231551038338166616, 8.05757434925996386695254234524, 9.211415543474982102109734803159, 10.80117234972604987699090166912, 12.02758146326727374823259185225, 12.53882557229182593148877058159, 14.63006540336975060339268219720, 15.069893861336696193667469869732, 16.7324528435714442003613825280, 17.44813818881389211339985288889, 18.53237544900074568817757871001, 19.350383578700706564675608936697, 20.19232582180585095960133372957, 21.107363431221526322277784761403, 23.142433093425407940854916061546, 23.979122651034534619502370648967, 24.62372247126644571972769609555, 25.55596801234553287199241253550, 26.67836784941171233217430250849, 27.7406749250959481412495169014, 28.3819906592907417510942660976

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