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Properties

Label 2-126-63.58-c1-0-5
Degree $2$
Conductor $126$
Sign $0.888 + 0.458i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
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  + 2-s − 1.73i·3-s + 4-s + (1.5 + 2.59i)5-s − 1.73i·6-s + (−2 − 1.73i)7-s + 8-s − 2.99·9-s + (1.5 + 2.59i)10-s + (1.5 − 2.59i)11-s − 1.73i·12-s + (−2.5 + 4.33i)13-s + (−2 − 1.73i)14-s + (4.5 − 2.59i)15-s + 16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.999i·3-s + 0.5·4-s + (0.670 + 1.16i)5-s − 0.707i·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s − 0.999·9-s + (0.474 + 0.821i)10-s + (0.452 − 0.783i)11-s − 0.499i·12-s + (−0.693 + 1.20i)13-s + (−0.534 − 0.462i)14-s + (1.16 − 0.670i)15-s + 0.250·16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50308 - 0.364499i\)
\(L(\frac12)\) \(\approx\) \(1.50308 - 0.364499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + 1.73iT \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.55531997267813121249278314769, −12.43664734168969743992896652938, −11.40599073417725308707227547191, −10.46822610720075562760876060459, −9.125235938597250740275394424889, −7.25688022136843343210620374945, −6.73568279451834012808834900322, −5.79569964434831973108258461910, −3.63371565836137938835886188259, −2.25602393735114106600285449273, 2.69731698787138918160492405238, 4.45103574423999985661409220879, 5.29691684266655132155212203925, 6.38984903996640421068437877837, 8.407100253915533330320315584223, 9.438248087041028602410420494484, 10.15311706372367480395942703027, 11.56789158902987700790310893324, 12.81902541248718095740813569189, 13.05462767031080919634940370834

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