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Properties

Label 2-126-63.38-c1-0-1
Degree $2$
Conductor $126$
Sign $-0.571 - 0.820i$
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  + i·2-s + (−0.290 + 1.70i)3-s − 4-s + (−0.0338 + 0.0585i)5-s + (−1.70 − 0.290i)6-s + (1.19 + 2.35i)7-s − i·8-s + (−2.83 − 0.993i)9-s + (−0.0585 − 0.0338i)10-s + (−3.40 + 1.96i)11-s + (0.290 − 1.70i)12-s + (3.32 − 1.92i)13-s + (−2.35 + 1.19i)14-s + (−0.0901 − 0.0747i)15-s + 16-s + (0.775 − 1.34i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.168 + 0.985i)3-s − 0.5·4-s + (−0.0151 + 0.0261i)5-s + (−0.697 − 0.118i)6-s + (0.452 + 0.891i)7-s − 0.353i·8-s + (−0.943 − 0.331i)9-s + (−0.0185 − 0.0106i)10-s + (−1.02 + 0.592i)11-s + (0.0840 − 0.492i)12-s + (0.922 − 0.532i)13-s + (−0.630 + 0.320i)14-s + (−0.0232 − 0.0193i)15-s + 0.250·16-s + (0.188 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ -0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.445890 + 0.853529i\)
\(L(\frac12)\) \(\approx\) \(0.445890 + 0.853529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.290 - 1.70i)T \)
7 \( 1 + (-1.19 - 2.35i)T \)
good5 \( 1 + (0.0338 - 0.0585i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.40 - 1.96i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.32 + 1.92i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.775 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.06 + 2.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.78 - 2.76i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.20 - 0.697i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (4.35 + 7.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.17 + 8.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.735 + 1.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.54T + 47T^{2} \)
53 \( 1 + (6.28 + 3.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.40T + 59T^{2} \)
61 \( 1 + 0.0815iT - 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 - 4.30iT - 71T^{2} \)
73 \( 1 + (-6.12 - 3.53i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.84T + 79T^{2} \)
83 \( 1 + (-3.93 + 6.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.84 - 10.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.363 + 0.209i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.96970513776151450868705088160, −12.82227156963710068950443628066, −11.53484144561815949874829257242, −10.56868650851627884172464915085, −9.356868313387623878334212920732, −8.562373333403859409694808377937, −7.27188475297218234475938633853, −5.51383766262963889596252147328, −5.06795572823937885440978158980, −3.21168646811924332821793682181, 1.23979253025057493175687068939, 3.15432523805280275370858037944, 4.94887783668567318055900263589, 6.40318992941408959752486656680, 7.79334922970673787647228034819, 8.558558727588256570780679747341, 10.25532431321037440435317167385, 11.09055091785929596212140962387, 11.92745264011252095593378823018, 13.14065833905847525960220346336

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