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Properties

Label 2-126-63.5-c1-0-4
Degree $2$
Conductor $126$
Sign $0.950 + 0.309i$
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 + (1.38 + 1.03i)3-s − 4-s + (0.714 + 1.23i)5-s + (1.03 − 1.38i)6-s + (0.327 − 2.62i)7-s + i·8-s + (0.843 + 2.87i)9-s + (1.23 − 0.714i)10-s + (2.96 + 1.70i)11-s + (−1.38 − 1.03i)12-s + (−5.48 − 3.16i)13-s + (−2.62 − 0.327i)14-s + (−0.294 + 2.45i)15-s + 16-s + (−1.14 − 1.97i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.800 + 0.599i)3-s − 0.5·4-s + (0.319 + 0.553i)5-s + (0.423 − 0.565i)6-s + (0.123 − 0.992i)7-s + 0.353i·8-s + (0.281 + 0.959i)9-s + (0.391 − 0.226i)10-s + (0.892 + 0.515i)11-s + (−0.400 − 0.299i)12-s + (−1.52 − 0.878i)13-s + (−0.701 − 0.0875i)14-s + (−0.0760 + 0.634i)15-s + 0.250·16-s + (−0.276 − 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.950 + 0.309i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.950 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29320 - 0.205259i\)
\(L(\frac12)\) \(\approx\) \(1.29320 - 0.205259i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 - 1.03i)T \)
7 \( 1 + (-0.327 + 2.62i)T \)
good5 \( 1 + (-0.714 - 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.96 - 1.70i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.48 + 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.14 + 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.97 - 4.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.298 - 0.172i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.34iT - 31T^{2} \)
37 \( 1 + (-1.07 + 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.202 + 0.350i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.51T + 47T^{2} \)
53 \( 1 + (-8.56 + 4.94i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 4.25T + 67T^{2} \)
71 \( 1 - 3.55iT - 71T^{2} \)
73 \( 1 + (0.201 - 0.116i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + (-0.811 - 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.02 - 3.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.47716565139074902684592740163, −12.31460103406589258403847338656, −11.01472636481579878525260039266, −10.03028623958248760566637852429, −9.624047838634852913926106124396, −8.087416318358813841685998382071, −6.99478460009262244241791987184, −4.89930494117606373198927786759, −3.75494401583534965368353577903, −2.32711345218800513281268296661, 2.14877369005035670743246459884, 4.23707726466484725178784444929, 5.83416741108511592212103562114, 6.87222458350524638756706118926, 8.216903073837322756319884014060, 8.960285452419834067424558152491, 9.740608408059920693112460381164, 11.87043295964180248795695892634, 12.49940091369341424882195617365, 13.60558185061287481387792813810

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