↑

Properties

Label 2-124-124.123-c1-0-6
Degree $2$
Conductor $124$
Sign $0.345 - 0.938i$
Analytic cond. $0.990144$
Root an. cond. $0.995060$
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  + (0.5 + 1.32i)2-s + 2.44·3-s + (−1.50 + 1.32i)4-s − 2·5-s + (1.22 + 3.24i)6-s + (−2.50 − 1.32i)8-s + 2.99·9-s + (−1 − 2.64i)10-s + 2.44·11-s + (−3.67 + 3.24i)12-s − 6.48i·13-s − 4.89·15-s + (0.500 − 3.96i)16-s + (1.49 + 3.96i)18-s + 5.29i·19-s + (3.00 − 2.64i)20-s + ⋯
L(s)  = 1  + (0.353 + 0.935i)2-s + 1.41·3-s + (−0.750 + 0.661i)4-s − 0.894·5-s + (0.499 + 1.32i)6-s + (−0.883 − 0.467i)8-s + 0.999·9-s + (−0.316 − 0.836i)10-s + 0.738·11-s + (−1.06 + 0.935i)12-s − 1.79i·13-s − 1.26·15-s + (0.125 − 0.992i)16-s + (0.353 + 0.935i)18-s + 1.21i·19-s + (0.670 − 0.591i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24640 + 0.869206i\)
\(L(\frac12)\) \(\approx\) \(1.24640 + 0.869206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 1.32i)T \)
31 \( 1 + (-4.89 + 2.64i)T \)
good3 \( 1 - 2.44T + 3T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 6.48iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 5.29iT - 19T^{2} \)
23 \( 1 + 4.89T + 23T^{2} \)
29 \( 1 - 6.48iT - 29T^{2} \)
37 \( 1 + 6.48iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 - 10.5iT - 47T^{2} \)
53 \( 1 - 6.48iT - 53T^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 - 6.48iT - 61T^{2} \)
67 \( 1 - 5.29iT - 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 9.79T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 12.9iT - 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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

−13.95957498629094910024103765499, −12.84937483242010727861195601035, −12.00395624162412182254853454332, −10.16284038462214387272404033523, −8.911280556195302179183540744744, −8.027897361289721953418470617625, −7.53822415158100126092679472078, −5.86057960904391230084136804222, −4.08166486813365887947808474250, −3.21295121914106049093678362386, 2.15147230775305375713916033678, 3.64945555047924801502731462516, 4.43805272419089999579219215565, 6.71851728792069102784705362380, 8.245571868945650299614471950669, 9.035523391642865048639896283270, 9.912990169974174223366195161330, 11.57363651114467325961571995140, 11.91835949702311914025024277671, 13.56091611181665441690516448532

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