Properties

Label 2-124-31.25-c3-0-2
Degree $2$
Conductor $124$
Sign $-0.600 - 0.799i$
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  + (4.00 + 6.93i)3-s + (−3.98 + 6.90i)5-s + (3.19 + 5.53i)7-s + (−18.5 + 32.1i)9-s + (2.85 − 4.95i)11-s + (−12.2 + 21.1i)13-s − 63.8·15-s + (−3.04 − 5.28i)17-s + (−26.3 − 45.6i)19-s + (−25.5 + 44.3i)21-s − 67.4·23-s + (30.7 + 53.2i)25-s − 81.1·27-s + 128.·29-s + (41.5 + 167. i)31-s + ⋯
L(s)  = 1  + (0.770 + 1.33i)3-s + (−0.356 + 0.617i)5-s + (0.172 + 0.298i)7-s + (−0.687 + 1.19i)9-s + (0.0783 − 0.135i)11-s + (−0.260 + 0.451i)13-s − 1.09·15-s + (−0.0435 − 0.0753i)17-s + (−0.318 − 0.551i)19-s + (−0.265 + 0.460i)21-s − 0.611·23-s + (0.245 + 0.425i)25-s − 0.578·27-s + 0.822·29-s + (0.240 + 0.970i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(7.31623\)
Root analytic conductor: \(2.70485\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :3/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.803673 + 1.60920i\)
\(L(\frac12)\) \(\approx\) \(0.803673 + 1.60920i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 + (-41.5 - 167. i)T \)
good3 \( 1 + (-4.00 - 6.93i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (3.98 - 6.90i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-3.19 - 5.53i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-2.85 + 4.95i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (12.2 - 21.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (3.04 + 5.28i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (26.3 + 45.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 67.4T + 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
37 \( 1 + (-119. - 206. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-4.18 + 7.24i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (0.491 + 0.851i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 513.T + 1.03e5T^{2} \)
53 \( 1 + (-302. + 524. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (229. + 397. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 - 16.2T + 2.26e5T^{2} \)
67 \( 1 + (58.8 - 102. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (238. - 413. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-498. + 862. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-528. - 914. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (128. - 222. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 393.T + 7.04e5T^{2} \)
97 \( 1 - 350.T + 9.12e5T^{2} \)
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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

−13.66265279933536820211497285793, −12.06821738563570573573270624546, −10.99687061016984084572990179864, −10.13042931020257952751607250768, −9.122682841801348849491820614255, −8.230005605177776503649461181079, −6.77389749578702078677034396148, −5.01623428591318209391099177683, −3.86122002095395105524031930338, −2.66828833044680183568084751702, 0.901475236808122727282522924743, 2.45479044751172857923500635965, 4.23656054743975398889359044326, 6.04542695258660716805145695008, 7.40451109139463556311704838221, 8.063791441973508253431936616727, 9.023875823752931441765321271087, 10.49131401589266373177090262460, 12.05304706767360304897457263193, 12.54499649257206924918947594141

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