↑

Properties

Label 2-931-133.132-c1-0-0
Degree $2$
Conductor $931$
Sign $-0.871 + 0.489i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.894i·2-s − 1.05·3-s + 1.20·4-s − 0.287i·5-s − 0.940i·6-s + 2.86i·8-s − 1.89·9-s + 0.257·10-s − 0.476·11-s − 1.26·12-s − 6.36·13-s + 0.302i·15-s − 0.157·16-s − 5.78i·17-s − 1.69i·18-s + (−2.59 + 3.50i)19-s + ⋯
L(s)  = 1  + 0.632i·2-s − 0.606·3-s + 0.600·4-s − 0.128i·5-s − 0.383i·6-s + 1.01i·8-s − 0.631·9-s + 0.0813·10-s − 0.143·11-s − 0.364·12-s − 1.76·13-s + 0.0781i·15-s − 0.0394·16-s − 1.40i·17-s − 0.399i·18-s + (−0.595 + 0.803i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.871 + 0.489i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.871 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0550502 - 0.210363i\)
\(L(\frac12)\) \(\approx\) \(0.0550502 - 0.210363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (2.59 - 3.50i)T \)
good2 \( 1 - 0.894iT - 2T^{2} \)
3 \( 1 + 1.05T + 3T^{2} \)
5 \( 1 + 0.287iT - 5T^{2} \)
11 \( 1 + 0.476T + 11T^{2} \)
13 \( 1 + 6.36T + 13T^{2} \)
17 \( 1 + 5.78iT - 17T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 - 5.71iT - 29T^{2} \)
31 \( 1 + 4.33T + 31T^{2} \)
37 \( 1 - 1.40iT - 37T^{2} \)
41 \( 1 + 7.44T + 41T^{2} \)
43 \( 1 + 7.20T + 43T^{2} \)
47 \( 1 - 4.13iT - 47T^{2} \)
53 \( 1 + 8.33iT - 53T^{2} \)
59 \( 1 - 1.92T + 59T^{2} \)
61 \( 1 - 2.62iT - 61T^{2} \)
67 \( 1 + 6.60iT - 67T^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + 1.31iT - 73T^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 - 0.627iT - 83T^{2} \)
89 \( 1 + 1.54T + 89T^{2} \)
97 \( 1 + 1.36T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.56455901239051493081618454265, −9.815369775321599525582438097516, −8.697011440947631415883374138380, −7.84959414146700856160220202722, −7.05075301418986316768618231155, −6.36998965727435509416250065138, −5.28635039014760374120837386767, −4.92810625753724392652822965698, −3.12153875039232460320146710266, −2.07774587348584033157311195662, 0.095919487739710152629054695479, 1.97821471000780248002219482169, 2.82344520509711336455611251075, 4.08020526729262558875929331105, 5.20957010647262877484893478314, 6.17828888697349954315016281952, 6.87978608342272967332190346296, 7.83389724416107525860630246370, 8.833086957147966259409503664443, 10.08495060560675883177077335594

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