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Properties

Label 2-124-4.3-c2-0-3
Degree $2$
Conductor $124$
Sign $0.364 - 0.931i$
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  + (−1.96 − 0.370i)2-s − 2.27i·3-s + (3.72 + 1.45i)4-s − 6.91·5-s + (−0.842 + 4.46i)6-s + 10.2i·7-s + (−6.78 − 4.24i)8-s + 3.83·9-s + (13.5 + 2.56i)10-s + 4.33i·11-s + (3.31 − 8.46i)12-s + 0.0177·13-s + (3.81 − 20.2i)14-s + 15.7i·15-s + (11.7 + 10.8i)16-s + 8.26·17-s + ⋯
L(s)  = 1  + (−0.982 − 0.185i)2-s − 0.757i·3-s + (0.931 + 0.364i)4-s − 1.38·5-s + (−0.140 + 0.744i)6-s + 1.46i·7-s + (−0.847 − 0.530i)8-s + 0.426·9-s + (1.35 + 0.256i)10-s + 0.394i·11-s + (0.276 − 0.705i)12-s + 0.00136·13-s + (0.272 − 1.44i)14-s + 1.04i·15-s + (0.734 + 0.678i)16-s + 0.486·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(3.37875\)
Root analytic conductor: \(1.83813\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 124,\ (\ :1),\ 0.364 - 0.931i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.451563 + 0.308205i\)
\(L(\frac12)\) \(\approx\) \(0.451563 + 0.308205i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 + 0.370i)T \)
31 \( 1 - 5.56iT \)
good3 \( 1 + 2.27iT - 9T^{2} \)
5 \( 1 + 6.91T + 25T^{2} \)
7 \( 1 - 10.2iT - 49T^{2} \)
11 \( 1 - 4.33iT - 121T^{2} \)
13 \( 1 - 0.0177T + 169T^{2} \)
17 \( 1 - 8.26T + 289T^{2} \)
19 \( 1 - 33.5iT - 361T^{2} \)
23 \( 1 - 30.2iT - 529T^{2} \)
29 \( 1 + 26.3T + 841T^{2} \)
37 \( 1 + 55.2T + 1.36e3T^{2} \)
41 \( 1 - 29.8T + 1.68e3T^{2} \)
43 \( 1 + 37.9iT - 1.84e3T^{2} \)
47 \( 1 - 24.7iT - 2.20e3T^{2} \)
53 \( 1 + 51.4T + 2.80e3T^{2} \)
59 \( 1 - 76.9iT - 3.48e3T^{2} \)
61 \( 1 - 93.3T + 3.72e3T^{2} \)
67 \( 1 + 16.2iT - 4.48e3T^{2} \)
71 \( 1 - 4.00iT - 5.04e3T^{2} \)
73 \( 1 + 115.T + 5.32e3T^{2} \)
79 \( 1 + 18.6iT - 6.24e3T^{2} \)
83 \( 1 + 82.5iT - 6.88e3T^{2} \)
89 \( 1 - 35.8T + 7.92e3T^{2} \)
97 \( 1 - 183.T + 9.40e3T^{2} \)
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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

−12.75642862255228467537355374743, −12.08421625099027750207921223870, −11.64713541754252658820710530133, −10.16317261246646745637822799302, −8.941738716869794516739666208005, −7.899949136442417904718962544715, −7.32321243440284294797679931901, −5.83229432302872009836810370500, −3.56877395133974238450718643662, −1.77497209174499462268709817840, 0.53164446066380415784108463077, 3.50958037716242248418960135827, 4.67851951488971717533510749627, 6.87503735909373825664727775985, 7.56084849300661404355045469282, 8.689196151876791313435941019937, 9.916620727788917898631469170135, 10.83433073309390733584832214815, 11.37090339202177855370255219143, 12.82672936066836401868593629948

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