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Properties

Label 2-124-124.119-c1-0-8
Degree $2$
Conductor $124$
Sign $0.0306 + 0.999i$
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  + (−1.32 + 0.482i)2-s + (0.532 − 0.921i)3-s + (1.53 − 1.28i)4-s + (−1.27 − 2.20i)5-s + (−0.262 + 1.48i)6-s + (−3.66 − 2.11i)7-s + (−1.41 + 2.44i)8-s + (0.933 + 1.61i)9-s + (2.76 + 2.32i)10-s + (−1.78 − 3.08i)11-s + (−0.367 − 2.09i)12-s + (4.37 − 2.52i)13-s + (5.89 + 1.04i)14-s − 2.71·15-s + (0.703 − 3.93i)16-s + (−0.0442 − 0.0255i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.341i)2-s + (0.307 − 0.532i)3-s + (0.766 − 0.641i)4-s + (−0.570 − 0.987i)5-s + (−0.107 + 0.605i)6-s + (−1.38 − 0.799i)7-s + (−0.501 + 0.865i)8-s + (0.311 + 0.539i)9-s + (0.873 + 0.733i)10-s + (−0.537 − 0.930i)11-s + (−0.106 − 0.605i)12-s + (1.21 − 0.700i)13-s + (1.57 + 0.278i)14-s − 0.700·15-s + (0.175 − 0.984i)16-s + (−0.0107 − 0.00619i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0306 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0306 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.0306 + 0.999i$
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: \((2,\ 124,\ (\ :1/2),\ 0.0306 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443061 - 0.429677i\)
\(L(\frac12)\) \(\approx\) \(0.443061 - 0.429677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.482i)T \)
31 \( 1 + (-5.53 + 0.568i)T \)
good3 \( 1 + (-0.532 + 0.921i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.27 + 2.20i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (3.66 + 2.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.78 + 3.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.37 + 2.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.0442 + 0.0255i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.11 - 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
29 \( 1 - 5.96iT - 29T^{2} \)
37 \( 1 + (5.83 + 3.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.69 + 2.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.73 - 4.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.85iT - 47T^{2} \)
53 \( 1 + (-9.86 + 5.69i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.83 - 1.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.374iT - 61T^{2} \)
67 \( 1 + (-5.80 + 3.34i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.04 + 3.48i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.10 - 1.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.71 - 4.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.32 - 4.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.64iT - 89T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−13.24126108950229838829732504657, −12.22276730529282988512546483695, −10.78600657513254467794512515965, −9.961664045624551849551704836296, −8.582701357002333697971632484456, −7.982776841254015566727415275968, −6.86238485936790613012711186087, −5.54578865876718100447117441352, −3.38576548307524699142972963804, −0.895807504838583617723585088536, 2.80087164304407830988839707101, 3.76063853092715995796581842045, 6.37320800756129772262129891742, 7.16038448347834606083838305882, 8.644432412859961394893737904434, 9.641953158252801886080901343194, 10.20445669951823790198245973550, 11.52631992756923844683805987385, 12.27812611987677646271809715269, 13.54247919622740101660131621146

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