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Properties

Label 2-124-124.123-c3-0-39
Degree $2$
Conductor $124$
Sign $-0.407 + 0.913i$
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  + (2.79 − 0.426i)2-s − 5.32·3-s + (7.63 − 2.38i)4-s − 4.56·5-s + (−14.8 + 2.27i)6-s − 24.7i·7-s + (20.3 − 9.92i)8-s + 1.39·9-s + (−12.7 + 1.94i)10-s − 27.3·11-s + (−40.6 + 12.7i)12-s − 37.1i·13-s + (−10.5 − 69.2i)14-s + 24.3·15-s + (52.6 − 36.4i)16-s − 119. i·17-s + ⋯
L(s)  = 1  + (0.988 − 0.150i)2-s − 1.02·3-s + (0.954 − 0.298i)4-s − 0.408·5-s + (−1.01 + 0.154i)6-s − 1.33i·7-s + (0.898 − 0.438i)8-s + 0.0517·9-s + (−0.403 + 0.0615i)10-s − 0.749·11-s + (−0.978 + 0.305i)12-s − 0.793i·13-s + (−0.201 − 1.32i)14-s + 0.418·15-s + (0.822 − 0.569i)16-s − 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.407 + 0.913i$
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: \((2,\ 124,\ (\ :3/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.839875 - 1.29399i\)
\(L(\frac12)\) \(\approx\) \(0.839875 - 1.29399i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.79 + 0.426i)T \)
31 \( 1 + (114. - 129. i)T \)
good3 \( 1 + 5.32T + 27T^{2} \)
5 \( 1 + 4.56T + 125T^{2} \)
7 \( 1 + 24.7iT - 343T^{2} \)
11 \( 1 + 27.3T + 1.33e3T^{2} \)
13 \( 1 + 37.1iT - 2.19e3T^{2} \)
17 \( 1 + 119. iT - 4.91e3T^{2} \)
19 \( 1 - 43.6iT - 6.85e3T^{2} \)
23 \( 1 - 65.5T + 1.21e4T^{2} \)
29 \( 1 - 27.2iT - 2.43e4T^{2} \)
37 \( 1 - 402. iT - 5.06e4T^{2} \)
41 \( 1 - 196.T + 6.89e4T^{2} \)
43 \( 1 - 463.T + 7.95e4T^{2} \)
47 \( 1 + 84.7iT - 1.03e5T^{2} \)
53 \( 1 + 410. iT - 1.48e5T^{2} \)
59 \( 1 + 551. iT - 2.05e5T^{2} \)
61 \( 1 - 30.9iT - 2.26e5T^{2} \)
67 \( 1 - 374. iT - 3.00e5T^{2} \)
71 \( 1 + 12.9iT - 3.57e5T^{2} \)
73 \( 1 + 718. iT - 3.89e5T^{2} \)
79 \( 1 + 54.3T + 4.93e5T^{2} \)
83 \( 1 + 566.T + 5.71e5T^{2} \)
89 \( 1 - 247. iT - 7.04e5T^{2} \)
97 \( 1 - 482.T + 9.12e5T^{2} \)
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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

−12.58676686165724803731015017526, −11.55822589768190140961074892289, −10.87313434363339394477967160314, −10.04827130838531992926964286702, −7.76589163827440796977061389908, −6.91267250202406631443656721664, −5.55026699254620854925594211283, −4.64952840162834017454750646395, −3.17083052968189282541849591164, −0.64842677983484269301534196792, 2.37557300405759578675210834957, 4.19790032990310587449815735037, 5.58324458529912252543759131663, 6.06662158039351058485400006044, 7.56166820320674543514187780974, 8.900207708032940854777012887553, 10.75843861958813424088644647247, 11.39041164126835891321493868240, 12.31998968558271217914532465013, 12.89157667831441263329073166431

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