Properties

Label 2-124-31.24-c2-0-1
Degree $2$
Conductor $124$
Sign $-0.342 - 0.939i$
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  + (0.938 + 4.41i)3-s + (−0.896 + 1.55i)5-s + (3.51 + 1.56i)7-s + (−10.3 + 4.62i)9-s + (−1.57 + 0.165i)11-s + (−10.8 + 9.72i)13-s + (−7.69 − 2.50i)15-s + (1.79 + 0.188i)17-s + (1.84 − 2.05i)19-s + (−3.60 + 16.9i)21-s + (20.8 − 28.6i)23-s + (10.8 + 18.8i)25-s + (−6.32 − 8.70i)27-s + (11.3 − 3.69i)29-s + (28.1 + 13.0i)31-s + ⋯
L(s)  = 1  + (0.312 + 1.47i)3-s + (−0.179 + 0.310i)5-s + (0.501 + 0.223i)7-s + (−1.15 + 0.514i)9-s + (−0.143 + 0.0150i)11-s + (−0.831 + 0.748i)13-s + (−0.513 − 0.166i)15-s + (0.105 + 0.0110i)17-s + (0.0972 − 0.108i)19-s + (−0.171 + 0.808i)21-s + (0.906 − 1.24i)23-s + (0.435 + 0.754i)25-s + (−0.234 − 0.322i)27-s + (0.392 − 0.127i)29-s + (0.907 + 0.419i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(3.37875\)
Root analytic conductor: \(1.83813\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.833186 + 1.18994i\)
\(L(\frac12)\) \(\approx\) \(0.833186 + 1.18994i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + (-28.1 - 13.0i)T \)
good3 \( 1 + (-0.938 - 4.41i)T + (-8.22 + 3.66i)T^{2} \)
5 \( 1 + (0.896 - 1.55i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-3.51 - 1.56i)T + (32.7 + 36.4i)T^{2} \)
11 \( 1 + (1.57 - 0.165i)T + (118. - 25.1i)T^{2} \)
13 \( 1 + (10.8 - 9.72i)T + (17.6 - 168. i)T^{2} \)
17 \( 1 + (-1.79 - 0.188i)T + (282. + 60.0i)T^{2} \)
19 \( 1 + (-1.84 + 2.05i)T + (-37.7 - 359. i)T^{2} \)
23 \( 1 + (-20.8 + 28.6i)T + (-163. - 503. i)T^{2} \)
29 \( 1 + (-11.3 + 3.69i)T + (680. - 494. i)T^{2} \)
37 \( 1 + (-32.9 + 19.0i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-20.1 - 4.27i)T + (1.53e3 + 683. i)T^{2} \)
43 \( 1 + (24.6 + 22.1i)T + (193. + 1.83e3i)T^{2} \)
47 \( 1 + (-25.5 + 78.5i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-3.08 - 6.92i)T + (-1.87e3 + 2.08e3i)T^{2} \)
59 \( 1 + (71.7 - 15.2i)T + (3.18e3 - 1.41e3i)T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 + (-0.643 + 1.11i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-40.2 + 17.9i)T + (3.37e3 - 3.74e3i)T^{2} \)
73 \( 1 + (-11.6 + 1.22i)T + (5.21e3 - 1.10e3i)T^{2} \)
79 \( 1 + (-116. - 12.2i)T + (6.10e3 + 1.29e3i)T^{2} \)
83 \( 1 + (10.0 - 47.1i)T + (-6.29e3 - 2.80e3i)T^{2} \)
89 \( 1 + (62.8 + 86.5i)T + (-2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (94.9 - 68.9i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−13.80323443305648020137194059049, −12.26430786524338445950970252181, −11.16964575587799221316872790068, −10.34141233546123039490135197440, −9.351798693592037538040818388614, −8.445695176414490438037114411681, −6.96369058028940742882212753125, −5.16844545791947282190606838150, −4.28000089852530814439903896208, −2.78187900479904302281579991979, 1.08618127343304623741479120538, 2.77122667777234631900766326621, 4.88604489971790909658552255793, 6.39399718383904926843938480576, 7.65893368934416683684631232382, 8.078199414881214470589402174788, 9.550004905219507177148896056650, 11.02799892258037412320611776703, 12.14656141136672462836785492029, 12.81444238139531486893630096925

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