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Properties

Label 2-124-124.119-c1-0-5
Degree $2$
Conductor $124$
Sign $0.947 - 0.318i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 1.03i)2-s + (−1.41 + 2.44i)3-s + (−0.136 − 1.99i)4-s + (0.855 + 1.48i)5-s + (1.16 + 3.81i)6-s + (4.34 + 2.51i)7-s + (−2.19 − 1.78i)8-s + (−2.47 − 4.29i)9-s + (2.35 + 0.546i)10-s + (−2.02 − 3.51i)11-s + (5.06 + 2.48i)12-s + (−0.936 + 0.540i)13-s + (6.79 − 2.07i)14-s − 4.82·15-s + (−3.96 + 0.544i)16-s + (−0.353 − 0.204i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + (−0.814 + 1.41i)3-s + (−0.0682 − 0.997i)4-s + (0.382 + 0.662i)5-s + (0.475 + 1.55i)6-s + (1.64 + 0.949i)7-s + (−0.775 − 0.631i)8-s + (−0.826 − 1.43i)9-s + (0.745 + 0.172i)10-s + (−0.611 − 1.05i)11-s + (1.46 + 0.716i)12-s + (−0.259 + 0.150i)13-s + (1.81 − 0.553i)14-s − 1.24·15-s + (−0.990 + 0.136i)16-s + (−0.0857 − 0.0495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27588 + 0.208864i\)
\(L(\frac12)\) \(\approx\) \(1.27588 + 0.208864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 1.03i)T \)
31 \( 1 + (-2.89 + 4.75i)T \)
good3 \( 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.855 - 1.48i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-4.34 - 2.51i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.02 + 3.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.936 - 0.540i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.353 + 0.204i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.51 + 0.877i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.521T + 23T^{2} \)
29 \( 1 + 4.98iT - 29T^{2} \)
37 \( 1 + (0.209 + 0.120i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 + 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.02 - 8.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.932iT - 47T^{2} \)
53 \( 1 + (-3.70 + 2.13i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.01 + 1.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + (-9.51 + 5.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.9 - 6.33i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.68 - 3.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0311 - 0.0538i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.12 - 1.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.18iT - 89T^{2} \)
97 \( 1 - 2.30T + 97T^{2} \)
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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

−13.58549600328619228204554449131, −11.93349379062845943059404965189, −11.27236589702631526101481537212, −10.72960134576581283610153167883, −9.750391917110306585177103134024, −8.482771304695702457221621883680, −6.07610727076365544303684183721, −5.29780715095233972430068446652, −4.36928923173104067676804489687, −2.61668820694002286795316635985, 1.72400083124685956821825648307, 4.71781750983202670099799205799, 5.35608225130793283539039394612, 6.89952993583553035586541079762, 7.56093559486346146391620233898, 8.477755976767710391752120837764, 10.62296779992569010858751271299, 11.75858749634738901996583765092, 12.55752069577464890684041361087, 13.29004125117257535197928061676

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