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Properties

Label 2-124-4.3-c2-0-24
Degree $2$
Conductor $124$
Sign $0.228 + 0.973i$
Analytic cond. $3.37875$
Root an. cond. $1.83813$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 − 0.229i)2-s − 5.14i·3-s + (3.89 − 0.913i)4-s + 2.15·5-s + (−1.18 − 10.2i)6-s + 9.67i·7-s + (7.52 − 2.71i)8-s − 17.4·9-s + (4.28 − 0.495i)10-s − 4.12i·11-s + (−4.69 − 20.0i)12-s − 15.9·13-s + (2.22 + 19.2i)14-s − 11.0i·15-s + (14.3 − 7.11i)16-s + 10.5·17-s + ⋯
L(s)  = 1  + (0.993 − 0.114i)2-s − 1.71i·3-s + (0.973 − 0.228i)4-s + 0.431·5-s + (−0.196 − 1.70i)6-s + 1.38i·7-s + (0.940 − 0.338i)8-s − 1.93·9-s + (0.428 − 0.0495i)10-s − 0.375i·11-s + (−0.391 − 1.66i)12-s − 1.22·13-s + (0.158 + 1.37i)14-s − 0.739i·15-s + (0.895 − 0.444i)16-s + 0.620·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.228 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.228 + 0.973i$
Analytic conductor: \(3.37875\)
Root analytic conductor: \(1.83813\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1),\ 0.228 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.96652 - 1.55860i\)
\(L(\frac12)\) \(\approx\) \(1.96652 - 1.55860i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.98 + 0.229i)T \)
31 \( 1 + 5.56iT \)
good3 \( 1 + 5.14iT - 9T^{2} \)
5 \( 1 - 2.15T + 25T^{2} \)
7 \( 1 - 9.67iT - 49T^{2} \)
11 \( 1 + 4.12iT - 121T^{2} \)
13 \( 1 + 15.9T + 169T^{2} \)
17 \( 1 - 10.5T + 289T^{2} \)
19 \( 1 - 11.6iT - 361T^{2} \)
23 \( 1 - 28.1iT - 529T^{2} \)
29 \( 1 - 43.3T + 841T^{2} \)
37 \( 1 - 47.7T + 1.36e3T^{2} \)
41 \( 1 + 64.6T + 1.68e3T^{2} \)
43 \( 1 + 65.2iT - 1.84e3T^{2} \)
47 \( 1 + 30.3iT - 2.20e3T^{2} \)
53 \( 1 + 69.7T + 2.80e3T^{2} \)
59 \( 1 - 1.18iT - 3.48e3T^{2} \)
61 \( 1 + 17.7T + 3.72e3T^{2} \)
67 \( 1 - 111. iT - 4.48e3T^{2} \)
71 \( 1 - 70.0iT - 5.04e3T^{2} \)
73 \( 1 + 44.6T + 5.32e3T^{2} \)
79 \( 1 + 85.5iT - 6.24e3T^{2} \)
83 \( 1 + 28.6iT - 6.88e3T^{2} \)
89 \( 1 + 62.9T + 7.92e3T^{2} \)
97 \( 1 - 41.1T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90933760538048086131050849642, −11.97388777574960454112557555534, −11.78871091598994326816637223144, −9.901673538257693460375542373259, −8.279345103342996248693959214493, −7.23689111679106486984737715594, −6.06324282907706513011009130780, −5.40372700128229306826584554810, −2.85536682929086121040365073312, −1.80577812684344056345439865015, 2.98343965380378021072746435462, 4.38953712185875479550070165170, 4.89024017655898401689905431950, 6.47372967461727162775859042402, 7.86967921849871205773407934540, 9.718861201223771666046164881579, 10.26614405187126607033617383142, 11.15380442760869897223916882766, 12.40848469802093616951347702433, 13.75102131966643079211854810982

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