Properties

Label 2-124-124.123-c3-0-7
Degree $2$
Conductor $124$
Sign $0.347 - 0.937i$
Analytic cond. $7.31623$
Root an. cond. $2.70485$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 2.78i)2-s + (−7.49 − 2.78i)4-s − 2·5-s + 33.4i·7-s + (−11.4 + 19.4i)8-s − 27·9-s + (−1 + 5.56i)10-s + (93 + 16.7i)14-s + (48.4 + 41.7i)16-s + (−13.5 + 75.1i)18-s + 55.6i·19-s + (14.9 + 5.56i)20-s − 121·25-s + (93 − 250. i)28-s + 172. i·31-s + (140. − 114. i)32-s + ⋯
L(s)  = 1  + (0.176 − 0.984i)2-s + (−0.937 − 0.347i)4-s − 0.178·5-s + 1.80i·7-s + (−0.508 + 0.861i)8-s − 9-s + (−0.0316 + 0.176i)10-s + (1.77 + 0.318i)14-s + (0.757 + 0.652i)16-s + (−0.176 + 0.984i)18-s + 0.672i·19-s + (0.167 + 0.0622i)20-s − 0.967·25-s + (0.627 − 1.69i)28-s + 0.999i·31-s + (0.776 − 0.630i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ 0.347 - 0.937i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.620947 + 0.431857i\)
\(L(\frac12)\) \(\approx\) \(0.620947 + 0.431857i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 2.78i)T \)
31 \( 1 - 172. iT \)
good3 \( 1 + 27T^{2} \)
5 \( 1 + 2T + 125T^{2} \)
7 \( 1 - 33.4iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 55.6iT - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 278T + 6.89e4T^{2} \)
43 \( 1 + 7.95e4T^{2} \)
47 \( 1 + 189. iT - 1.03e5T^{2} \)
53 \( 1 - 1.48e5T^{2} \)
59 \( 1 + 523. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 857. iT - 3.00e5T^{2} \)
71 \( 1 - 656. iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 1.90e3T + 9.12e5T^{2} \)
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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

−12.81268344637674264412799311812, −11.92391377500608265827532219908, −11.45203557702265082189986049585, −10.05350103809888349052676677392, −8.923107405462924579964981770043, −8.291986757122217922277701682317, −6.00192615884594482724549271996, −5.16300131950600311541676092085, −3.33298338377701237097363509113, −2.12503024523908125777662830582, 0.36425813691943199516077323340, 3.54136928556152791095773913448, 4.69108481642176139809910298238, 6.15548640948323978985853591114, 7.29244739467627077681133455726, 8.098104056701677226481838952610, 9.394121043452456148714338570571, 10.58056503746539082675882648508, 11.74354484781918643953353949918, 13.24965469543222111953890737899

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