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Properties

Label 2-124-124.27-c1-0-4
Degree $2$
Conductor $124$
Sign $0.617 - 0.786i$
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  + (−0.642 + 1.26i)2-s + (0.224 − 0.690i)3-s + (−1.17 − 1.61i)4-s + 0.618·5-s + (0.726 + 0.726i)6-s + (2.48 + 3.42i)7-s + (2.79 − 0.442i)8-s + (2 + 1.45i)9-s + (−0.396 + 0.778i)10-s + (1.17 − 0.854i)11-s + (−1.38 + 0.449i)12-s + (−4.73 − 1.53i)13-s + (−5.91 + 0.937i)14-s + (0.138 − 0.427i)15-s + (−1.23 + 3.80i)16-s + (4.04 − 5.56i)17-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)2-s + (0.129 − 0.398i)3-s + (−0.587 − 0.809i)4-s + 0.276·5-s + (0.296 + 0.296i)6-s + (0.941 + 1.29i)7-s + (0.987 − 0.156i)8-s + (0.666 + 0.484i)9-s + (−0.125 + 0.246i)10-s + (0.354 − 0.257i)11-s + (−0.398 + 0.129i)12-s + (−1.31 − 0.426i)13-s + (−1.58 + 0.250i)14-s + (0.0358 − 0.110i)15-s + (−0.309 + 0.951i)16-s + (0.981 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(0.990144\)
Root analytic conductor: \(0.995060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866732 + 0.421217i\)
\(L(\frac12)\) \(\approx\) \(0.866732 + 0.421217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 1.26i)T \)
31 \( 1 + (2.85 + 4.78i)T \)
good3 \( 1 + (-0.224 + 0.690i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 0.618T + 5T^{2} \)
7 \( 1 + (-2.48 - 3.42i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (-1.17 + 0.854i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (4.73 + 1.53i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.04 + 5.56i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.75 - 1.54i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.48 - 1.80i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (3.19 - 1.03i)T + (23.4 - 17.0i)T^{2} \)
37 \( 1 - 2.17iT - 37T^{2} \)
41 \( 1 + (0.236 + 0.726i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.31 + 4.04i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.21 + 1.04i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.28 + 1.76i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.30 + 1.07i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 - 9.95iT - 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 + (2.57 - 3.54i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.690 + 0.951i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-11.8 - 8.61i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.98 + 6.11i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (10.1 + 13.9i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.04 + 4.39i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.81944977264807914849215884581, −12.64799776672334401830662438145, −11.55474433149578914800184610067, −10.09697520042633973201687680491, −9.176142865918865812734627955260, −8.024096227692592565340866570396, −7.27148706768193625656570811040, −5.72935621483113726394901771416, −4.87841020033375319656909219101, −2.02530258016256654434371266501, 1.66045752618035996595193436747, 3.84661500504135174801916353175, 4.67689258372987025437537345968, 7.01909470834615920022217773213, 8.035148324629014993470713911341, 9.388959269542027635622368490246, 10.21209064035623398466813738605, 10.91851151462895895899444681866, 12.19603986886308610596936676000, 13.02280712812241700148416983142

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