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Properties

Label 1-131-131.9-r0-0-0
Degree $1$
Conductor $131$
Sign $0.939 + 0.343i$
Analytic cond. $0.608361$
Root an. cond. $0.608361$
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  + (0.779 + 0.626i)2-s + (0.0241 − 0.999i)3-s + (0.215 + 0.976i)4-s + (0.958 − 0.285i)5-s + (0.644 − 0.764i)6-s + (−0.527 + 0.849i)7-s + (−0.443 + 0.896i)8-s + (−0.998 − 0.0483i)9-s + (0.926 + 0.377i)10-s + (0.981 + 0.192i)11-s + (0.981 − 0.192i)12-s + (0.926 − 0.377i)13-s + (−0.943 + 0.331i)14-s + (−0.262 − 0.964i)15-s + (−0.906 + 0.421i)16-s + (−0.681 − 0.732i)17-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (0.0241 − 0.999i)3-s + (0.215 + 0.976i)4-s + (0.958 − 0.285i)5-s + (0.644 − 0.764i)6-s + (−0.527 + 0.849i)7-s + (−0.443 + 0.896i)8-s + (−0.998 − 0.0483i)9-s + (0.926 + 0.377i)10-s + (0.981 + 0.192i)11-s + (0.981 − 0.192i)12-s + (0.926 − 0.377i)13-s + (−0.943 + 0.331i)14-s + (−0.262 − 0.964i)15-s + (−0.906 + 0.421i)16-s + (−0.681 − 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(131\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(0.608361\)
Root analytic conductor: \(0.608361\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{131} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 131,\ (0:\ ),\ 0.939 + 0.343i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.724853145 + 0.3053223787i\)
\(L(\frac12)\) \(\approx\) \(1.724853145 + 0.3053223787i\)
\(L(1)\) \(\approx\) \(1.620303603 + 0.2238332150i\)
\(L(1)\) \(\approx\) \(1.620303603 + 0.2238332150i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad131 \( 1 \)
good2 \( 1 + (0.779 + 0.626i)T \)
3 \( 1 + (0.0241 - 0.999i)T \)
5 \( 1 + (0.958 - 0.285i)T \)
7 \( 1 + (-0.527 + 0.849i)T \)
11 \( 1 + (0.981 + 0.192i)T \)
13 \( 1 + (0.926 - 0.377i)T \)
17 \( 1 + (-0.681 - 0.732i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 + (-0.989 + 0.144i)T \)
29 \( 1 + (-0.998 + 0.0483i)T \)
31 \( 1 + (0.715 + 0.698i)T \)
37 \( 1 + (-0.861 + 0.506i)T \)
41 \( 1 + (-0.906 - 0.421i)T \)
43 \( 1 + (-0.607 + 0.794i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.485 - 0.873i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.607 - 0.794i)T \)
71 \( 1 + (0.885 - 0.464i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.568 + 0.822i)T \)
83 \( 1 + (0.399 + 0.916i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.0724 - 0.997i)T \)
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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

−28.70512445938553485010909029668, −27.91718704325484052262875603956, −26.61591187388228446252413942414, −25.78327005541698607524247191629, −24.569622158988475647207515205583, −23.18218329697599077356883053240, −22.379225209360363635606566276685, −21.695009093567815499294465383905, −20.691802888691587131991005499722, −19.976328963048080982742619200551, −18.7309795958970980215880240465, −17.18592546466317863182135170452, −16.26624283151667297859024127440, −14.961133518097051647881776839778, −14.006740228425159967700512557792, −13.373299584669178566291022787130, −11.75551697451238116080462232419, −10.618548569990445557184535065780, −10.000057361666457316860339492371, −8.93363473750856809521641982441, −6.511004427040851986806644674844, −5.78942956054583621344877548422, −4.147166279926208089749728603903, −3.499637830363875713780245541170, −1.77889120004681452850703329455, 1.9180836117967313361147569091, 3.17273802702869487231877753652, 5.11578328909159478494770352343, 6.203929358439241395797615959649, 6.776836556239701887929759990038, 8.431901224605204496553769915971, 9.26520810915066283041884551479, 11.4462327173007122814773427882, 12.395926966105796915026707330651, 13.35110255420966182746188285497, 13.944665563142283470403368901893, 15.26095586577761285631998139468, 16.43308033667059443301079993577, 17.6103778318355800422381125375, 18.19126040586868798926346806378, 19.74329012121280842081507746740, 20.80609298458748698804187730134, 22.15666179314871296587262528342, 22.59539952951740943798627632033, 24.032220584607603868997067991860, 24.74913294055258502331025159707, 25.44356269243826906777536202390, 26.08175627325643881022595722018, 28.02657877522964511242497063365, 28.98552996578558297833956879271

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